text:
# If and NDSolve Causing Issues [duplicate]

I have the code below, which simulates a particle of mass $m$ in a fourth order polynomial potential given an initial conditions $q_0,p_0$. That is to say, I am simulating Hamilton's equations using NDSolve.

params = {m -> 1.47*^-25, f0 -> 100*^3, ff -> -100*^3};
tf = 200*^-6;
tffreq = tf; (*Final time for ramping of frequency*)
V = α*q^2 + β*q^4 + γ*q;
b = 1*^-4 &;
g = 1*^-23 &;

f = ff + (f0 - ff)*(tffreq - #)/tffreq /. params &;
at = Sign[f[#]]*.5*m*(2*Pi*f[#])^2 /. params &;
a = If[#<tffreq,at[#],at[tffreq]]&;

q0 = -1.72092*^-10;
gradV = D[V, q] /. {γ -> #1, α -> #2, β -> #3,
q -> q[t]} &;

{qout, pout} = {q, p} /.
NDSolve[{Evaluate[D[q[t], t] == p[t]/m /. params],
D[p[t], t] == -1*gradV[g[t], a[t], b[t]], q[0] == q0,
p[0] == 0}, {q, p}, {t, 0, tf}][[1]];
Plot[qout[t], {t, 0, tf}]

Which gives an incorrect solution for qout. However, if I change

at = Sign[f[#]]*.5*m*(2*Pi*f[#])^2 /. params &;
a = If[#<tffreq,at[#],at[tffreq]]&;

to

a = Sign[f[#]]*.5*m*(2*Pi*f[#])^2 /. params &;

I find the qout that I expect. When I plot at[t] and a[t] I find that they give the correct results, so I would think that NDSolve would behave identically in both instances. Additionally, if I change the conditioning in the If to If[1>0,...], I also get the correct results.

For reference, the reason I separate tf and tffreq is because after a certain time (tffreq) I would like a[t] to be a constant. My guess is that it is a precision error somewhere, but then I don't know why it would still be plotting the correct a[t] even when I condition it.

Here is the correct simulation result, without the conditioning:

Here is the incorrect simulation result, with the If conditioning:


text:
# Where is the right triangle in sides a, b, c, for a hyperbola?

Homework Helper
Gold Member
An ellipse has some model standard form values, a, b, and c which are easily enough to identify from the graph and parts of the graph related to the ellipse's graph. Seeing the right triangle relating a, b, and c, is easy enough. The Pythagorean Theorem is used to relate these three values.

I have been hoping to find and SEE the values of a, b, and c in their right triangle for a hyperbola, but I just cannot find this triangle. I have looked in a couple of College Algebra textbooks, and the most that is done is to assign a substitution as part of the derivation of the equation for the hyperbola. That just does not help to show where to find the right triangle. $$a^2+b^2=c^2$$ comes simply from the part of $$c^2-a^2$$
but no picture to support it.

SteamKing
Staff Emeritus
Homework Helper
An ellipse has some model standard form values, a, b, and c which are easily enough to identify from the graph and parts of the graph related to the ellipse's graph. Seeing the right triangle relating a, b, and c, is easy enough. The Pythagorean Theorem is used to relate these three values.

I have been hoping to find and SEE the values of a, b, and c in their right triangle for a hyperbola, but I just cannot find this triangle. I have looked in a couple of College Algebra textbooks, and the most that is done is to assign a substitution as part of the derivation of the equation for the hyperbola. That just does not help to show where to find the right triangle. $$a^2+b^2=c^2$$ comes simply from the part of $$c^2-a^2$$
but no picture to support it.
The right triangle is located between the center of the hyperbola, the vertex, and the point above the vertex lying on the asymptote.

https://mysite.du.edu/~jcalvert/math/hyperb.htm

If we have a hyperbola whose center C is at the origin, the vertices V1 and V2 will be located at the points ($\pm$a, 0), while the asymptotes have the equations y = $\pm$(b/a)x. When x = a, at the positive vertex V1, y = b, and the relation

c$^{2}$ = a$^{2}$ + b$^{2}$ is also satisfied.

Homework Helper
Gold Member
Hi SteamKing,

I looked at and read some of the article. Nice, but a little difficult to follow. I still do not see/find a clear way to understand thinking from a and c, to the meaning and value of b. I can well accept that the b value and placement is realistic and meaningful; but I have trouble thinking through a good way to reach b from the hyperbola and a and c.

I have the right feeling that a rectangle must be associated with a hyperbola because this would correspond to what occurs in an ellipse. The ellipse has a, b, an c, easy to see. Then, one could expect to find somewhere, a meaningful rectangle(in a hyperbola). The rotation (for the hyperbola) of the segment from center to either focus (the one on the right) then will meet the asymtpote, but then this seems strange to understand. I become lost at that. I have no way to think why to make this rotation. The asymptote and slope of the asymptote then seems to involve b, but then the b is what I am trying to understand. I could go on and just use b on faith and could answer some common academic questions about hyperbolas at the "Algebra 2" and "College Algebra" level, but I was just hoping to get inside the meaning and not need to use just faith. I feel like this lack of understanding about a hyperbola will be an obstacle to leaning conic sections and more about coordinate geometry and much more about Mathematics.

Last edited:
SteamKing
Staff Emeritus
Homework Helper
I don't know why you are dragging 'faith' into this situation.

As for an ellipse, if a is the length of the semi-major axis and b is the length of the semi-minor axis, then c is the distance between these two points. It's not clear why this is significant to you or to the construction of an arbitrary ellipse.

The equation of the asymptotes to the hyperbola are as stated. They are straight lines. I don't understand why this is difficult to grasp.

Homework Helper

format: plain


text:
# Which principle of sports rule provides the opportunity for equal competition for everyone?

###### Question:

Which principle of sports rule provides the opportunity for equal competition for everyone?

### Which of the following is not a physical factor that constitutes a region? Climate Religion River Systems Vegetation

Which of the following is not a physical factor that constitutes a region? Climate Religion River Systems Vegetation...

### 11^2^3x=983298f+0220.2^2(38) whats x?

11^2^3x=983298f+0220.2^2(38) whats x?...

### It is -15° in Bismarck and -20° in Calgary. How much colder is Calgary than Bismarck?

It is -15° in Bismarck and -20° in Calgary. How much colder is Calgary than Bismarck?...

### Can you download 12 new songs to your MP3 player then you delete five old songs write each amount as an integer ￼PLEASE

Can you download 12 new songs to your MP3 player then you delete five old songs write each amount as an integer ￼PLEASE...

### 14. Nutrient-dense foods are better than foods that are less nutritious because

14. Nutrient-dense foods are better than foods that are less nutritious because...

### If a ship is sailing on an ocean and it is 2:00pm at the prime meridian and 10:00 am on the ship what is the longitudinal location of the ship

If a ship is sailing on an ocean and it is 2:00pm at the prime meridian and 10:00 am on the ship what is the longitudinal location of the ship...

### Iron reacts with Oxygen gas to form Iron II Oxide according to the reaction below. 4 Fe + 3 02 2 Fe2O3 How many moles of Iron II Oxide can be produced from 347.7 L of Oxygen gas at a temperature of 74.3 °C and a pressure of 294.5 kPa? Round Answers to 0.01 decimals

Iron reacts with Oxygen gas to form Iron II Oxide according to the reaction below. 4 Fe + 3 02 2 Fe2O3 How many moles of Iron II Oxide can be produced from 347.7 L of Oxygen gas at a temperature of 74.3 °C and a pressure of 294.5 kPa? Round Answers to 0.01 decimals...

### What was the effect of Japanese imperialism on the development of nationalism in the early 1900s? A. Nationalism developed very slowly. B. Imperialism brought other cultures to the Japanese and nationalism waned.. . C. Nationalism became an important part of Japanese culture. D. Imperialism caused the Japanese to lose interest in nationalism as they explored other parts of the world.

What was the effect of Japanese imperialism on the development of nationalism in the early 1900s? A. Nationalism developed very slowly. B. Imperialism brought other cultures to the Japanese and nationalism waned.. . C. Nationalism became an important part of Japanese culture. D...

### What is the acceleration of a projectile at its highest point? What is the acceleration just before and after the point?

What is the acceleration of a projectile at its highest point? What is the acceleration just before and after the point?...

### Find the range of the following piecewise function. [0,11) (0,11] [-4,6) (-4,6]

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### Jazakallah khair allah hu akbar

jazakallah khair allah hu akbar...

### All 7th grade students are ______ on the first day of school. They just want to make new friends and see their teachers again

All 7th grade students are ______ on the first day of school. They just want to make new friends and see their teachers again...

### In what pattern does the flow of energy in the ecosystem take place? multirectional bidrectional unidirectional

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### * The first Portuguese captain to reach India and return was Christopher Columbus. Juan de Pizzaro. Vasco da Gama. Bartolomeu Dias.

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### Finally, your teacher would like to know: What is one thing you learned in Banzai that you didn't know before?

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### How is the motif of weather significant to the events? Also, consider the symbol of water (61). Juxtapose the beauty that Victor sees in the tempest, with the horror he feels at the sight of “the wretch” (62).

How is the motif of weather significant to the events? Also, consider the symbol of water (61). Juxtapose the beauty that Victor sees in the tempest, with the horror he feels at the sight of “the wretch” (62)....

### What is the answer to this question

What is the answer to this question...


text:
# Why is a QM wave function not normalizable

In quantum mechanics, why is a wave function not normalizable if it always has the same sign as its second derivative?

Meir Achuz
Homework Helper
Gold Member
In 1D, if f"/f is positive, the function curves away from the x axis.
To be normalizable, it must in some region curve toward the x axis.

dextercioby
Homework Helper
In quantum mechanics, why is a wave function not normalizable if it always has the same sign as its second derivative?

Your question doesn't make (too much) sense, since a wave function is a complex (generally with a nonzero imaginary part) valued function and to say that $a+ib >0$ makes no sense.


text:
# What are standard controllers for automatic pH control?

My background in control theory comes from electrical engineering (robots, motors, etc.) and I want to build a control system for automatic pH control.

Can I directly apply my knowledge from classical control to chemical control? E.g. can I use a PID controller in a similar way as I am use to, or are there special pitfalls/best practices in chemistry that I should be aware of?

Short system description: My plan is to have two pumps that can add either base or acid to a solution under constant agitation (shaker plate), using a standard glass electrode sensor for measurements.

• Welcome to Chemistry.SE! Please take a minute to check out the help center and tour page to better understand our guidelines and question policies. – A.K. Sep 3 '18 at 16:57
• Certainly. Thermometer and heater, or pHmeter and a pump each for acid and base, the difference is just that you can fire in two directions in the latter case, which should make the control circuit easier to balance. – Karl Sep 3 '18 at 16:58
• I do have 2 pumps, one for base and one for acid – Toke Faurby Sep 4 '18 at 16:58

Automatic pH control is not that easy as is shown by the fact that automatic titrators, the equipment that you want to build, are relatively expensive. Besides titration, automatic titrators usually have also a pH stat function, which is for maintaining a certain fixed pH.

In first place, note that to have a good control, typically you must know which type of system you are working with.

Normally, to maintain a constant pH only one reagent is added, either acid or base. This is because chemical systems typically become more acidic or more alkaline when they proceed, but do not oscillate. When you add both reagents, they usually start to compete: a bit of acid is added, it passes the set point, base is added, it passes the set point in the other direction, etc. This happens especially when the response of the system is slow and the mixing is not good (shaker plate is not a good idea, use a stirrer instead). The way to solve this is by adjusting very well the concentrations of the reagents, the delay times, addition rates, etc., and this is different in every system.

You also should know in which part of the pH curve you are working to set the concentration of the reagents and the rate of addition. The pH curve has three very different zones, a vertical part and two horizontal parts.

Depending on where you are, the conditions must be different. Note also that the inflection point becomes less obvious when weak acids or bases are involved.

Finally, you might need to integrate also a temperature compensation since, depending on the $\Delta\epsilon$ you have set, the pH change with temperature will be to high.

In short, a PID might work in some specific system, but for a general purpose system you will probably need some more sophisticated control like a PLC.


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Are numbers sets?

One of the milestones of contemporary philosophy of mathematics is Benacerraf‘s 1965 article “What Numbers Could Not Be“.1 There he offers a compelling argument according to which numbers cannot be considered as sets — namely, they cannot be metaphysically identified with sets.

The argument is quite simple: if numbers were sets, we should be able to find a unique progression of sets with which numbers can be identified. But this is apparently impossible: there is a lot of ω-series that can serve as well for the aim. For example, we can adopt von Neumann’s series, and say that $0=\emptyset, 1=\{\emptyset\}, 2=\{\emptyset,\{\emptyset\}\}$, and so on, where the successor function $S$ is defined by $S(x)=x \cup \{x\}$. Or we can adopt Zermelo’s series, and say that $0=\emptyset, 1=\{\emptyset\}, 2=\{\{\emptyset\}\}$, and so on, where the successor function $S$ is defined by $S(x)=\{x\}$. Now, the problem is: is $3=\{\{\{\emptyset\}\}\}$ or is $3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$? Benacerraf presents then the example of two children, Ernie and John. The first learned that von Neumann’s ordinals are the natural numbers, while the latter that Zermelo’s ordinals are the natural numbers. Now, they will be easily able to learn arithmetic set theoretically via the above constructions, and they will agree on any arithmetical theorem, except that for Ernie it is true that $3 \in 17$, while for John it is false!

It can sound quite odd to ask whether a number belongs or not to another number. It is actually not an arithmetical question. But this is the point: Ernie and John agree on every arithmetical question; they disagree only on non-arithmetical issues, but these issues cannot be considered as essential in order to point out the metaphysical status of numbers. We can do arithmetic either with Zermelo’s ordinals or von Neumann’s ordinals; but if we are going to metaphysically identify numbers with sets, we must choose one of the two. We cannot admit that $\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}=\{\{\{\emptyset\}\}\}$. But how can we choose? According to Benacerraf, we cannot choose, since there is no arithmetical reason to prefer one series on the other. Thus, for we can’t have more than one set of natural numbers, we must admit that numbers are not sets.

But then? What are numbers? Benacerraf’s solution is that we have to move from objects to structures. What permits to Ernie and John to agree on arithmetical theorems is not the nature of any single number, but the fact that they are considering two different instantiations of the very same structure — so we can say that numbers are anything that has the right kind of structure.

Benacerraf’s article has been variously discussed over the years but there have been very few attempts to directly challenge his argument. One of the most interesting of these attempts can be found in Eric Steinhart’s 2002 article “Why Numbers are Sets”.2 In this article, Steinhart argues that we actually have reasons to prefer von Neumann’s ordinals over Zermelo’s ordinals (and also over any other alternative solution), and hence we can overcome Benacerraf’s worries and say that yes, numbers are really (metaphysically) sets. Benacerraf poses two conditions on a set-theoretic structure $\alpha= (\omega,f,e,\prec)$ in order to be a good candidate for being the natural numbers:3

• arithmetical condition (AC): $\alpha$ satisfies the AC iff $(\omega,f,e)$ is a model of the Dedekind-Peano Axioms;
• cardinal condition (CC): $\alpha$ satisfies the CC iff it identifies the numerical ‘less than’ relation $<$ with the set-theoretic relation $\prec$ such that the cardinality of a set $S$ is $n$ iff there is a 1-1 correspondence between $S$ and $\{m|n \prec m\}$.

Taken together, AC and CC constitutes the “Natural Number condition” (NNC). According to Benacerraf, NNC suffices to define the natural numbers, so that any other condition we want to eventually add is just over the necessary — that’s what makes the choice between Zermelo’s and von Neumann’s ordinals impossible. Steinhart, on the contrary, claims that actually

there is one set of sets that stands out very clearly for the mathematicians as the natural numbers. The mathematicians standardly identify the natural numbers with the finite von Neumann ordinals. They make the identification because not all apparent reductions satisfy the NN-conditions equally well. (p. 345)

Five reasons, according to Steinhart, justify this preference:

1. the set of von Neumann ordinals is recursively defined;
2. its sets uniquely satisfies certain ordering conditions;
3. it is uniformly extendible to the transfinite ;
4. it is a minimal $\omega$-series;
5. its $n$-th member is the set of all $m$ less than $n$.

1-5 actually specifies five further conditions, beyond NNC, that $\alpha$ has to satisfy in order to be the natural numbers. However, one might still hold that, even if 1-5 justify the mathematicians’ preference for von Neumann ordinals, they still don’t compel us to admit that numbers have to be metaphysically identified with this set of ordinals. Indeed, 1-5 seem to be rather stylistical reasons and seem not to have a metaphysical relevance for our choice. Yet, what is interesting in Steinhart’s article is a mathematical proof he gives to convince us that actually natural numbers are von Neumann ordinals. The proof runs as follows.

CC implies the following axiom (C1) and definition (C2):

• C1: for all $x$, if $x$ is in $\omega$, then there exists a set $x^*=\{z \in \omega | z;
• C2: the cardinality of any set $S$ is $x$ iff there exists some 1-1 correspondence between $x$ and $x^*$.

Now, let’s suppose we choose an $\alpha=(\omega,f,e,\prec)$ to serve as our natural numbers. Because of CC, we must admit that, for each $\alpha$-number $\alpha_n$, the set of $\alpha$-numbers less than $\alpha_n$ is in the NN-universe (i.e., it exists). Thus, if we assume that $\alpha$ is the natural numbers, we must admit that our NN-universe contains the cardinality sets $\{\emptyset\}, \{\alpha_0\}, \{\alpha_0, \alpha_1\},\{\alpha_0, \alpha_1, \alpha_2\}$ and so on. Hence, the NNC entails that $\omega^* = \{x^*|x \in \omega\}$ is in the NN-universe. Similarly, we can define $f^*, e^*$ and $\prec^*$, and show that they are all in our NN-universe. So, we can prove that if $\alpha=(\omega,f,e,\prec)$ satisfies the NNC, then $\alpha^*=(\omega^*,f^*,e^*,\prec^*)$ is in the NN-universe and satisfies the NNC. Therefore, if $\alpha=N$, then $\alpha$ satisfies the NNC; but if $\alpha$ satisfies the NNC, then $\alpha^*$ satisfies the NNC, and hence we can say that $\alpha^*=N$. It follows that $\alpha= \alpha^*$. This means that $\omega=\omega^*$, and therefore that $n=n^*$ for any $n$ in $\omega$. But if $n=n^*$ for any $n$ in $\omega$, then $0=\emptyset$ and $n=\{0,1,2,\ldots,n-1\}=\{n-1 \cup \{n-1\}\}$, and these are precisely the von Neumann ordinals. Then, if $\alpha$ is the natural numbers, then $\alpha$ is the von Neumann finite ordinals.

The argument is very interesting, but I think it is not valid. I am not completely sure about it, so I will present my objection very cautiously. The key point in Steinhart argument consists in the fact that the C1 imposes the existence, for any $x$ in $\omega$, of the set $x^*=\{z \in \omega | z \prec x\}$. According to Steinhart,

If the NN-conditions assert some rule $R$, then the NN-universe contains the domain of $R$, the range of $R$, the extension of $R$, and nothing else. For if we cannot reason to the existence of these objects in the NN-universe, then that rule is meaningless (it plays no role in determining the models of the NN-conditions). (p.353)

This is what warrants Steinhart in saying that $\omega^*$ is in the NN-universe, and that’s precisely what I can’t understand. It seems to me that there is no need to say that the NN-universe has to contain all this stuff. Of course, the domain, the range and the extension of $R$ must be contained in the universe of our set theory (the set theory in which we carry on the reduction), but there’s no need to say that they have to be contained in the NN-universe. Now, if I am right in noticing this, then we are no longer entitled to say that $\omega^*$ is in the NN-universe. It is in the universe of our set theory, and, since it satisfies NNC, this means that we have a different way to identify numbers with sets; but we can no longer conclude that $\alpha=\alpha^*$, and hence the proof is no longer valid. Simply, $\alpha$ implies the existence in our set-theory universe of $\alpha^*$, which is needed in order to say that the cardinality of a set $S$ is $n$ iff $S$ can be put in a 1-1 correspondence with the set $n^*=\{m|m \prec n\}$. It remains a matter of style whether we want adopt this or that progression of sets, but nothing compels us to say that if we can identify numbers with sets, then we cannot identify numbers with anything but von Neumann finite ordinals.

A further consideration can be made, as a conclusion of this post. If I am right in rejecting Steinhart’s proof (and I am not sure I am right in doing this — I restate), then the only way to reject Benacerraf’s conclusion is to hold a strong version of mathematical naturalism: numbers are von Neumann finite ordinals because that’s what working mathematicians do. And, viceversa, if we agree with Benacerraf, we cannot stick ourselves to a too much strong version of mathematical naturalism.

[1] P. Benacerraf (1965), “What Numbers Could Not Be”, The Philosophical Review, vol. 74(1), pp. 47-73. Reprinted in P. Benacerraf and H. Putnam (eds) (1984), Philosophy of Mathematics, Cambridge University Press, New York, pp. 272-295.
[2] E. Steinhart (2002), “Why Numbers Are Sets”, Synthese, vol. 133(3), pp. 343-361.
[3] $\omega$ is a set of sets, $f$ is a one-to-one function from $\omega$ to $\omega$ , $e$ is a particular set belonging to $\omega$, $\prec$ is a set-theoretic relation. The idea is that if we identify $\alpha$ with the natural numbers, we have that $\omega$ is the set of natural numbers, $f$ is the successor function, $e$ is the initial number $0$ and $\prec$ is the ‘less than’ relation.


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# 0.1 Review exercises (ch 3-13)  (Page 11/12)

 Page 11 / 12

130. Out of the years with water rationing, find the probability that there is a drought.

Use the following information to answer the next three exercises:

Apple Pumpkin Pecan
Female 40 10 30
Male 20 30 10

131. Suppose that one individual is randomly chosen. Find the probability that the person’s favorite pie is apple or the person is male.

132. Suppose that one male is randomly chosen. Find the probability his favorite pie is pecan.

133. Conduct a hypothesis test to determine if favorite pie type and gender are independent.

Use the following information to answer the next two exercises: Let’s say that the probability that an adult watches the news at least once per week is 0.60.

134. We randomly survey 14 people. On average, how many people do we expect to watch the news at least once per week?

135. We randomly survey 14 people. Of interest is the number that watch the news at least once per week. State the distribution of X . X ~ _____

136. The following histogram is most likely to be a result of sampling from which distribution?

1. Chi-Square
2. Geometric
3. Uniform
4. Binomial

137. The ages of De Anza evening students is known to be normally distributed with a population mean of 40 and a population standard deviation of six. A sample of six De Anza evening students reported their ages (in years) as: 28; 35; 47; 45; 30; 50. Find the probability that the mean of six ages of randomly chosen students is less than 35 years. Hint: Find the sample mean.

138. A math exam was given to all the fifth grade children attending Country School. Two random samples of scores were taken. The null hypothesis is that the mean math scores for boys and girls in fifth grade are the same. Conduct a hypothesis test.

n $\overline{x}$ s 2
Boys 55 82 29
Girls 60 86 46

139. In a survey of 80 males, 55 had played an organized sport growing up. Of the 70 females surveyed, 25 had played an organized sport growing up. We are interested in whether the proportion for males is higher than the proportion for females. Conduct a hypothesis test.

140. Which of the following is preferable when designing a hypothesis test?

1. Maximize α and minimize β
2. Minimize α and maximize β
3. Maximize α and β
4. Minimize α and β

Use the following information to answer the next three exercises: 120 people were surveyed as to their favorite beverage (non-alcoholic). The results are below.

Beverage/Age 0–9 10–19 20–29 30+ Totals
Milk 14 10 6 0 30
Soda 3 8 26 15 52
Juice 7 12 12 7 38
Totals 24 330 44 22 120

141. Are the events of milk and 30+:

142. Suppose that one person is randomly chosen. Find the probability that person is 10–19 given that he or she prefers juice.

143. Are “Preferred Beverage” and “Age” independent events? Conduct a hypothesis test.

144. Given the following histogram, which distribution is the data most likely to come from?

1. uniform
2. exponential
3. normal
4. chi-square

## Chapter 3

1. c. parameter

2. a. population

3. b. statistic

4. d. sample

5. e. variable

6. quantitative continuous

7.

1. 2.27
2. 3.04
3. –1, 4, 4

## Chapter 4

9. c. (0.80)(0.30)

10. b. No, and they are not mutually exclusive either.

How do you get log of normal population
plz give me answer of this question show that mean of population or sampling distribution are equally
Mari
yes
Hazrat
what is the probability of getting no head face up in three tosses of a fair coin?
In how many ways can probability be assigned to an event of interest?
Epara
Hey guys can someone help me with combinations and permutations
Lion
Phone lines on an airline system are occupied 50% of time assume that 10 calls are placed to the airline. What is the probability at least 1 call the lines are occupied?
Phone lines on an airline system are occupied 50% of time assume that 10 calls are placed to the airline. What is the probability at least 1 call the lines are occupied?
Why is the method of selecting the sample even more important than the size of the sample?
formulaas for gruoped and ungrouped of quartiles
Why is the method of selecting the sample even more important than the size of the sample?
Nana
una empresa productora que participa en el mercado de lapices tiene la siguente funcion de demanda Qd=a la raiz de 9 -9p si la elasticidad precio de demanda de la empresa es epd=0,5 determinar a) hallar el precio y cantidad
find the mean mew and the standard devation sigma of the given population 9.8,10.2,10.4,9.8,10.0,10.2,9.6
1. A card is drawn at random from an ordinary deck of 52 playing cards.* *Find the probability that it is* (a). an ace (b). a jack of hearts (c). a three of clubs or a six of diamonds (d). a heart (e). any suit except hearts (f). a ten or a spade (g). neither a four nor a club. *Hint:* 1st determine how many of the following is in a deck of cards. A deck of cards have 52 cards ♠- Club ♣ - Spade ♥ - Heart ♦- Diamond
a measure of central tendency is a typical value around which other figures congregate
hmm mean mode median
Umar
nothing
farri
Y = alpha0 + alpha1X1 + E what is this equation
An estimated linear regression equation
Carlos
Thank you
Musawenkosi
simple linear regression .. where Alpho zero is reg constant ( intercept of the reg line) and Alpha1 is the regression coefficient ( slope of the regression line)
Umar
null and altarnate hypothesis are the statement about
Umar
don't know
farri
I said we give hypothesis about any population and mean , in null hyp we say sample mean is equal to the population mean where as in alternate hyp we say sample mean is not equal to the pop mean .. to test these things we use students test statistics commonly
Umar
ok thnx
farri
welcome
Umar
probability of getting one black card
from a standard deck?
Umar
it will be 1/2
Umar
coz there are 26 blacks card out of 52 in a deck
Umar
so prob of getting a black card out of the deck = 26/52 = 1/5 =0.5
Umar
when are not two events mutually exclusive? 1) they overlap in a venn diagram 2) Probability of one affects the other
both
Umar
A stenographer claims that she can take dictation at the rate of 120 words per minute can we reject her claim on the basis of 100 trails in which she demonstrates a mean of 116 words with a variance of 225 words
the hypothesis to be tested is the claim to be tested
smita
H0= 120,Ha not equal to 120 x bar =116,s=225,n=100
smita
how to solve probability questions


text:
lilypond-user
[Top][All Lists]

## Re: invisible rest that takes no horizontal space

 From: Peter Van Kranenburg Subject: Re: invisible rest that takes no horizontal space Date: Fri, 18 Jun 2010 12:23:11 +0200 User-agent: Mozilla/5.0 (Macintosh; U; Intel Mac OS X 10.6; en-US; rv:1.9.1.9) Gecko/20100317 Thunderbird/3.0.4

On 6/16/10 1:33 PM, Kieren MacMillan wrote:

Hi Peter,

adding *0 to s[0-9]+ is easy. To add the \partial command, it is necessary to
figure out the current time signature from the context, to find the start of
the measure and to figure out the length of the partial bar, which is not easy.
One has to write a lilypond parser for that.

No... if you have

s4 s4 s4. c8 |

you could easily have your script figure out the length of the measure, without a
"Lilypond parser".
Of course, I'm assuming you use barline checks, like the docs recommend --
otherwise, you may definitely be unable to use this solution.

These partial measures can occur anywhere. Not only at te beginning (see attached example). And, no, there are no barline checks encoded.

The example has been encoded as (starting at second system):

[...] d4. e8 fis4 e d2 s2 \bar ":|:" fis2 g [...]

So, my question is how to prevent the s2 from occupying horizontal space.

best,
Peter

example.png
Description: PNG image


text:
Thread: Non 15k to 1M anyone? View Single Post
 2004-03-07, 13:01 #5 Thomas11     Feb 2003 27·3·5 Posts Hi Joss and Masser, thank's for your interest. First of all I should note that I'm currently out of town for about one week. Therefore I haven't full access to all the data I'd like to provide to you. For the Nash weight computation I used a small C program written by myself after a "close" look into John Brennen's Java applet. I developed my program in a UNIX environment using the GMP library. I compiled and used it under Linux too, but I don't know if GMP is available for Windows and how to get it compiled there. Nevertheless, if some of you are interested, I could send you the source code and/or Linux binary. Please don't ask for a "complete" list of Nash weights for all (odd) k's up to 350 million. Since my program stores the results into ascii files this would be 3.5 GByte of data (10.5 MByte per 1 million range of k's = 500.000 odd k's)! But any small portion of data can be easily extracted for the list, e.g. the "less than 100" list or a "larger than 6000" list or a "zero weight" list (= Riesel numbers). The "less than 100" list would be about 900 kByte (or 300 kByte ZIPed). Originally I had planned to take one or a few k's up to 10M and I did a lot of sieving work in that direction. But after all it is too much work for one single person having only a few machines. So any of you may take part in the search for some big primes ... The following 23 k values have been sieved for n=2..10.000.000 up to p=8.000.000.000.000 (yes, 8 trillion!) and already LLR-tested up to n=1.000.000: Code:  k w w' prime for n= ----------------------------------------- 19370947 17 15 25 59910449 15 15 92 80857169 19 17 4 143316643 15 16 162405629 20 21 896, 12236 175437131 20 21 189030223 18 19 203012861 16 17 754 209826493 16 18 224371169 17 17 3548 243163663 15 18 919087 245265883 16 17 260213857 13 16 265831619 19 16 276278983 16 18 21623, 473423 290851087 23 21 57 298095191 14 15 300871183 22 23 308120317 19 19 308141737 19 19 7517 315940139 18 17 8388, 595620 326840893 17 15 31 ---------------------------------------- (w is the "original" Nash weight for n=100.001-110.000, while w' is the Nash weight obtained for n=1-10.000) I have no idea, which k should be choosen for testing n>1 million. Maybe one or a few for which no prime exists for n<1 million. As I already mentioned, I don't have access to the presieved ranges at the moment, but I can send them to Joss around March 15th, so he could store the data on 15k.org. It should be mentioned that a single LLR test on a n=1 million number takes about 4 hours on a 2.4 GHz P4. And around n=1.5 million the time raises to 8 hours and more. In case you can't wait until I've sent you the presieved data and/or you want to test some untested k-values for lower n, here is a short list of low weight k's which I haven't tested so far (but someone else may have tested them): Code:  k w w' ---------------------- 10013593 40 46 10247561 38 38 10284899 27 30 10346561 46 48 10453199 28 26 10463923 34 28 10598947 45 45 10639619 37 38 10671431 34 36 10805593 49 45 10813783 34 40 10906603 38 38 10932097 44 41 10943321 50 61 11223059 26 27 11311003 31 28 11319193 45 43 11468609 46 47 11639819 39 38 11658187 48 44 11716993 48 41 11741347 42 44 11846279 50 48 11847299 45 45 11932211 38 37 11955659 36 36 12254533 36 36 12305981 44 44 12334093 49 53 12551291 43 41 12695159 31 36 12753311 48 42 12825793 49 53 13277471 48 50 13651423 48 49 13768087 46 49 13900807 23 18 14085941 35 36 14321533 29 23 14466883 40 41 14533213 42 44 14549347 48 52 14745343 31 32 14961487 25 23 ---------------------- I suggest to sieve them for n=2..250.000 up to about p=10 or 15 billion. Only 150-400 candidates will remain per k and can be LLR tested in a few hours. Candidates for n<256 should be tested by Pari (or Mathematica, Maple, etc.) rather than using LLR, since it may crash or report a prime as "non-prime". There should be quite a few primes in the above list for n<250.000 but only one or two Top5000 primes. Best wishes and a few primes ... Thomas


text:
# Interpreting effects of categorical and continuous predictors in multiple linear mixed models

I'm building a LMM model with a continuous DV (signal amplitude) and two IV: one continuous (questionnaire score) and one categorical with 3 levels (condition). Conditions are: S, M and SM (the last being a combination of the first two). They are dummy coded and SM is set as the reference level. I use one random effect - a random intercept for subjects. Each subject encountered all the conditions. This is the model in R's lme4:

m = lmer(DV ~ Q.score * cond + (1|subj), data = df, REML = FALSE)

(I read that with balanced designs and only one random effect I can use ML instead of REML.)

lmerTest::summary() gives these estimates of fixed effects: (According to the documentation, the t- and p- values are calculated with Satterthwaite approximation for degrees of freedom)

    Fixed effects:
Estimate Std. Error         df t value    Pr(>|t|)
(Intercept)        -0.941813   0.179603 103.900000  -5.244 0.000000832 ***
Q.score            -0.043102   0.024048 103.900000  -1.792       0.076 .
condSMvsM           0.752574   0.179470 104.000000   4.193 0.000057923 ***
condSMvsS           0.385522   0.179470 104.000000   2.148       0.034 *
Q.score:condSMvsM   0.005157   0.024030 104.000000   0.215       0.830
Q.score:condSMvsS   0.024290   0.024030 104.000000   1.011       0.314

Since cond is a multilevel categorical predictor, it's easier to interpret the results looking at the whole effect of the predictor. I know there are many ways of doing that so I tried a few, but I am surprised by the differences between them, especially in the p-values for the interaction (short comment: I know p-values aren't the miraculous answers to what is or is not true, but I simply have to get them, as my boss requires so, so please take a deep breath with me and be so kind to help anyway...).

1) Analysis of variance table of type II (or whatever you specify as the SS type) with Satterthwaite approximation for degrees of freedom - lmerTest::anova

              Sum Sq Mean Sq NumDF DenDF F.value    Pr(>F)
Q.score       2.4049  2.4049     1    52  2.8717 0.0961287 .
cond         14.7285  7.3643     2   104  8.7937 0.0002961 ***
Q.score:cond  0.9501  0.4750     2   104  0.5672 0.5688346

2) Analysis of Deviance Table (Type II Wald chisquare tests) - car::Anova

               Chisq Df Pr(>Chisq)
Q.score       2.8717  1  0.0901471 .
cond         17.5875  2  0.0001517 ***
Q.score:cond  1.1345  2  0.5670901

3) The likelihood ratio test - with lmer models

> m0 = lmer(DV ~ 1 + (1|subj), data=df, REML = FALSE)
> m1 = lmer(DV ~ Q.score + (1|subj), data=df, REML = FALSE)
> m2 = lmer(DV ~ cond + (1|subj), data=df, REML = FALSE)
> m3 = lmer(DV ~ Q.score * cond + (1|subj), data=df, REML = FALSE)
> anova(m0,m1,m2,m3)
Data: df
Models:
object: DV ~ 1 + (1 | subj)
..1: DV ~ Q.score + (1 | subj)
..2: DV ~ cond + (1 | subj)
..3: DV ~ Q.score * cond + (1 | subj)
Df    AIC    BIC  logLik deviance   Chisq Chi Df Pr(>Chisq)
object  3 513.25 522.40 -253.62   507.25
..1     4 512.45 524.65 -252.23   504.45  2.7953      1  0.0945439 .
..2     5 501.16 516.41 -245.58   491.16 13.2916      1  0.0002666 ***
..3     8 503.24 527.64 -243.62   487.24  3.9236      3  0.2698349

And here is my first question: (1) Which test do I use to report an effect fo a categorical predictor? Should it be F-value from analysis of variance? Or chi square statistics (if so, from the analysis of deviance or from the likelihood ratio test)?

The effects don't differ very much in my example, but I am still facing another problem:

(2) How to describe the direction of an effect of a multilevel categorical predictor?

I have seen publications in which authors report coefficients, t-values and SD of fixed effects. If I understand correctly, I could do the same from the summary() output (see the first piece of code). However, since I can only report pairwise comparisons between levels, I would have to change contrasts to address the pair of levels which are not the reference level in my first model - here: S vs. M. (I don't have an easy way of deciding on the contrasts (like clinical group vs. control group), but three arbitrary conditions). For example, this is summary for the same model with condition M as the reference value (as compared to SM above):

Fixed effects:
Estimate Std. Error         df t value  Pr(>|t|)
(Intercept)        -0.189239   0.179603 103.900000  -1.054    0.2945
Q.score            -0.037945   0.024048 103.900000  -1.578    0.1176
condMvsS           -0.367052   0.179470 104.000000  -2.045    0.0434 *
condMvsSM          -0.752574   0.179470 104.000000  -4.193 0.0000579 ***
Q.score:condMvsS    0.019133   0.024030 104.000000   0.796    0.4277
Q.score:condMvsSM  -0.005157   0.024030 104.000000  -0.215    0.8305

And here I have two problems:

(3) Why does the continuous predictor's coefficient and t value change with different contrasts of the categorical predictor? The Q.score t and p values changed. Of course the overall effect of this predictor stays the same (as found by one of the above methods). Does is mean that the value of Q.score decreases more when "going" from the M than SM condition? Should I then only report the general effect of this predictor (like in the above methods)?

(4) Shouldn't I apply some kind of correction for multiple comparisons if I change the contrasts to explore all pairwise comparisons? Can I do it at all?...

And one more general question in the end:

(5) How to acquire a general F statistic of the whole model? Or is R^2 and adj. R^2 enough?

I know it's a lot for one post, but I really read everything I could find and I'm still stuck. I would really, really appreciate your help with any of these questions!

• Since your questions are almost identical to the issues I have bumped into, I was wondering if you have gotten any insights since you have posted this question? Thank you so much! Jan 15, 2021 at 11:05


text:
Integration using Partial Fractions

1. Feb 27, 2006

Hootenanny

Staff Emeritus
I need to find the following intergral:
$$\int_{0}^{1} \frac{28x^2}{(2x+1)(3-x)} \;\; dx$$
So I split it into partial fractions thus:
$$\frac{2}{2x+1} + \frac{36}{3-x} - 14$$
Then integrated:
$$\int_{0}^{1} \frac{2}{2x+1} + \frac{36}{3-x} - 14 \;\; dx$$
$$= \left[ \ln\left| 2x+1 \right| + 12\ln\left| 2-x \right| - 14x \right]_{0}^{1}$$
But this isn't going to give me the correct answer which is quoted as:
$$37\ln 3 - 36\ln 2 - 14$$
Can anybody see where I've gone wrong? Thank's

2. Feb 27, 2006

Hootenanny

Staff Emeritus
Sorry just a correction in my answer:
$$= \left[ \ln\left| 2x+1 \right| + 12\ln\left| 3-x \right| - 14x \right]_{0}^{1}$$

3. Feb 27, 2006

Physics Monkey

Your partial fraction expansion is ok. However, you integrated $$\frac{36}{3 - x}$$ incorrectly.

4. Feb 27, 2006

Hootenanny

Staff Emeritus
Ahh yes. Thank-you, I realise what I've done now.

5. Feb 27, 2006

Hootenanny

Staff Emeritus
Is there any standard intergral in the form of $\int \frac{a}{b+x} = ...$ because at the moment I'm not sure what to do. If anybody could guide me through it?

6. Feb 27, 2006

Kamataat

I suppose you can use $$\int\frac{a}{b\pm x}dx=\pm a\ln|x\pm b|$$

- Kamataat

7. Feb 27, 2006

Hootenanny

Staff Emeritus
Thank you very much.

8. Dec 29, 2007

jellybeanzgir

hi there, I was wondering if you could possibly help me with an integration problem

$$\int_{1/4}^{-1/4} \frac{1}{(1-4x^2)} \;\; dx$$

i managed to split it into partial fractions and hence split up the integration as follows

$$\frac{1}{2} \int_{1/4}^{-1/4} \frac{1}{2x+1} - \frac{1}{2x-1} \;\;dx$$

after this i integrated it in the following manner but im unsure as to whether it is right, and if so, how to continue it.

$$[\frac{1}{2} (\frac{1}{2}\ln(2x+1)-\frac{1}{2}\ln(2x-1))]$$still between the same two limits

Can anyone offer a hand?

9. Dec 29, 2007

jellybeanzgir

if my integration was correct please continue reading this post, if not then it doesn't matter!!

I broke down that inegraton to read

$$[\frac{1}{2} \ln\frac{2x+1}{2x-1}]$$ limits $$\pm\frac{1}{4}$$

i then simplified this down to the following (including limit values)

$$\frac{1}{2} \ln\frac{1.5}{-0.5} - \frac{1}{2}\ln\frac{0.5}{-1.5}$$

Using the rules of logs i calculated this to be

$$\ln\frac{-3}{\frac{-1}{3}}$$

Giving a final answer of $$\ln9$$

Does anyone know if this is the correct answer or have i gone about it in the wrong way?

10. Dec 29, 2007

Rainbow Child

The correct answer is

$$\frac{1}{4} \left(\ln|2x+1|-\ln|-2x+1|\right)$$

you need the absolute values in $$\ln(x)$$ in order to be well defined.

See Kamataat's post $$\int\frac{a}{b\pm x}dx=\pm a\ln|x\pm b|$$

11. Dec 29, 2007

rocomath

well you have $$\frac{1}{1-4x^2}$$

so it should become

$$\frac{1}{(1+2x)(1-2x)}$$

if you want to write it as (2x+1)(2x-1), then you have to factor out a negative

12. Dec 29, 2007

rocomath

you have it the other way around, i even checked it on my calculator and my answer matches:

after integrating it should become

$$\frac{1}{4}\left[\ln\left|\frac{1-2x}{1+2x}\right|\right]_{.25}^{-.25}$$

13. Dec 29, 2007

jellybeanzgir

thanks

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text:
# 5 biocViews

Packages added to the Bioconductor Project require a biocViews: field in their DESCRIPTION file. The field name “biocViews” is case-sensitive and must begin with a lower-case ‘b.’

biocViews terms are “keywords” used to describe a given package. They are broadly divided into three categories, representing the type of packages present in the Bioconductor Project -

1. Software
2. Annotation Data
3. Experiment Data

biocViews are available for the release and devel branches of Bioconductor. The devel branch has a check box under the tree structure which, when checked, displays biocViews that are defined but not used by any package, in addition to biocViews that are in use.

## 5.1 Motivation

One can use biocViews for two broad purposes.

1. A researcher might want to identify all packages in the Bioconductor Project which are related to a specific purpose. For example, one may want to look for all packages related to “Copy Number Variants.”

2. During development, a package contributor can “tag” their package with biocViews so that when someone looking for packages (like in scenario 1) can easily find their package.

## 5.2 biocViews during new package development

Visit the devel branch of biocViews when you are in the process of adding biocViews to your new package. Identify as many terms as appropriate from the hierarchy. Prefer ‘leaf’ terms at the end of the hierarchy, over more inclusive terms. Remember to check the box displaying all available terms.

1. Your package will belong to only one part of Bioconductor Project (Software, Annotation Data or Experiment Data), so choose only only biocViews from that category.

2. biocViews listed in your package must match exactly (e.g., spelling, capitalization) the terms in the biocViews hierarchy.

When you submit your new package for review , your package is checked and built by the Bioconductor Project. We check the following for biocViews:

1. Package contributor has added biocViews.

2. biocViews are valid.

3. Package contributor has added biocViews from only one of the categories.


text:
# Math Help - Integral calculation

1. ## Integral calculation

Hi! I have a problem solving this integral.Could someone help me or give me an advice?

$
$
$\int_{0}^{2\pi} cos(\theta)exp(a \cos(\theta))d\theta
$

2. Originally Posted by hokutose
Hi! I have a problem solving this integral.Could someone help me or give me an advice?

$
$
$\int_{0}^{2\pi} cos(\theta)exp(a \cos(\theta))d\theta
$

Are you sure you copied down the integral correctly?

Mathematica is telling me this can't be solved exactly...

$


text:
# Homework Help: Covariance of Discrete Random Variables

1. Apr 8, 2009

### Shackman

1. The problem statement, all variables and given/known data
Find E(XY), Cov(X,Y) and correlation(X,Y) for the random variables X, Y whose joint distribution is given by the following table.

X
1 2 3
Y -1| 0 .1 .1

0| 0 .5 .6

1| .2 0 0

3. The attempt at a solution

The covariance and correlation fall into place quite easily once I have found E(XY). I have found E(X), E(Y), Var(X) and Var(Y) but none of these values help as the variables are not independent. So in trying to find E(XY), I am trying to set up the double integral, but am confused by the fact that the variables are discrete. The limits of integration are not obvious and it is not obvious how to integrate a discrete function either. Is there another way I can look at this?

Last edited: Apr 8, 2009
2. Apr 8, 2009

### Billy Bob

$$E[XY]=\sum\sum xyf(x,y)$$

3. Apr 9, 2009

### Shackman

That is so much better. Thanks. The organization of my book is pretty terrible, I'm just finding that equation now.


text:
##### Review badges
0 pre-pub reviews
0 post-pub reviews
##### Abstract

Van der Waals forces are determinants of the formation of protein-ligand complexes. Physical models based on the Lennard-Jones potential can estimate van der Waals interactions with considerable accuracy and with a computational complexity that allows its application to molecular docking simulations and virtual screening of large databases of small organic molecules. Several empirical scoring functions used to evaluate protein-ligand interactions approximate van der Waals interactions with the Lennard-Jones potential. In this chapter, we present the main concepts necessary to understand van der Waals interactions relevant to molecular recognition of a ligand by the binding pocket of a protein target. We describe the Lennard-Jones potential and its application to calculate potential energy for an ensemble of structures to highlight the main features related to the importance of this interaction for binding affinity.

##### Authors

Bitencourt-Ferreira, Gabriela;  Veit-Acosta, Martina;  de Azevedo, Walter Filgueira, Jr.

• 2 authors


text:
Electrostatics Questions

The electric field is defined as the electric force per unit charge. The electric field is directed radially outward from a positive charge and inside towards a negative point charge. A static charge can generate only an electric field in the surrounding space. The electromagnetic field is a combination of both electric and magnetic fields. Stationary charges produce the electric field, and moving charges produce the magnetic field. These are the sources of the electromagnetic field.

Electrostatics is the branch of physics that deals with properties of charges at rest and slow-moving electric charges. When charged particles are transferred from one body to another, we can experience the existence of static electricity. One of the most common examples of the electrostatic force is sticking plastic wrap to one’s hand after it is removed from a package.

This concept arises from the forces applied by the electric charges and is explained by Coulomb’s law.

The formula gives coulomb’s Law of electrostatics:

The electric force on a charge Q1 under these conditions, due to a charge Q2 at a distance r, is given by Coulomb’s law:

$$\begin{array}{l}F=k\frac{q_{1}q_{2}}{d^{2}}\end{array}$$

Electric field (\vec{E})is a vector quantity defined as the electrostatic force \vec{E} in newtons on a hypothetical small test charge at the point due to Coulomb’s Law, divided by the magnitude of the charge q in coulombs

\vec{E}=\frac{\vec{F}}{q}

Electric field lines start on a positive charge and end on a negative charge. The density of electric field lines gives the amount (magnitude) of the electric field at any given point.

The smallest unit charge is 1.6021 x 10^-19 Coulomb (C). Neutral particles contain the same amount of electrons and protons.

Gauss’ law states that “In any closed surface of any shape placed in free space, the total electric flux in an electric field is proportional to the total electric charge enclosed by the surface”.

$$\begin{array}{l}\oint_{s}^{} \vec{E}.\vec{dA} = \frac{1}{\varepsilon_{0}}Q_{enclosed}=\int_{v}^{}\frac{\rho}{\varepsilon _{0}}.d^{3}r \end{array}$$

Gauss’s Law in differential form:

$$\begin{array}{l}\vec{\bigtriangledown} \vec{E} = \frac{\rho }{\varepsilon_{0}}\end{array}$$

Poisson and Laplace equations

$$\begin{array}{l}{\bigtriangledown}^{2} \phi = -\frac{\rho}{\varepsilon_{0}}\end{array}$$

This relationship is a form of Poisson’s equation. When there is no unpaired electric charge, the equation becomes Laplace’s equation:

$$\begin{array}{l}{\bigtriangledown}^{2} \phi = 0\end{array}$$

Coulomb force is also called electrostatic force or Coulomb interaction. It is the force of attraction or repulsion of particles that arises due to the electric charges of particles. The force created by the electric field is much stronger than the force created by the magnetic field.

1. Define electrostatics.

Electrostatics is the branch of physics that deals with phenomena and properties of stationary or slow-moving electric charges.

2. State Coulomb’s Law.

The force of attraction or repulsion between two charged bodies is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

3. What is the value of the electron (e) charge?

1. 1.60217662 × 10^-29
2. 1.60217662 × 10^-27
3. 1.60217662 × 10-9
4. 1.60217662 × 10-19

Explanation: The value of electron charge is 1.60217662 × 10^-19.

4. Neutral particles contain _____.

1. Equal numbers of protons and electrons
2. More protons than electrons
3. More electrons than protons
4. None of the options

Answer: a) Equal numbers of protons and electrons.

5. Which of the following is an example of electrostatics?

1. Shining of star
2. The attraction of paper to a charged scale
3. The explosion of grain silos
4. Option b) and c)

Answer: d) Option b) and c)

6. The electrostatic force is also known as the _____.

1. Charge force
2. Coulomb force
3. Electron force
4. None of the option

7. Coulomb’s law is given by the formula

1. $$\begin{array}{l}F=k\frac{q_{1}+q_{2}}{d^{2}}\end{array}$$
2. $$\begin{array}{l}F=k\frac{q_{1}-q_{2}}{d^{2}}\end{array}$$
3. $$\begin{array}{l}F=k\frac{q_{1}q_{2}}{d^{2}}\end{array}$$
4. $$\begin{array}{l}F=k\frac{2*q_{1}q_{2}}{d^{2}}\end{array}$$

8. Electric field is a _____.

1. Scalar quantity
2. Vector quantity
3. Dimensionless quantity
4. None of the options

Explanation: Electric field is a quantity that has both magnitude and direction.

9. In Coulomb’s law, ε is known as ____._

1. Relative permittivity
2. The permittivity of free space
3. Absolute permittivity
4. None of the option

Explanation: Absolute permittivity is a measure of the electric polarizability of a dielectric.

10. Define Static electricity.

Static electricity is an electrical event where charged particles are transferred from one body to another.

Practice Questions

1. Explain Gauss Law in brief.
2. Give some examples to explain the electrostatic force.
3. List some properties of the electric field.
4. Give the formula of Coulomb’s law.
5. State true or false: Coulomb force is the attraction or repulsion of particles or objects because of their electric charge.

Watch the video below to understand electrostatics better.


text:
Let Q = L a K f 3 . Suppose the firm seeks to produce a given output while minimizing its total

Let Q = L a K f 3 . Suppose the firm seeks to produce a given output while minimizing its total input cost: TC = P L L + P K K. Show that the optimal quantities of labor and capital satisfy L/K = (a/f3)(P K /P L ). Provide an intuitive explanation for this result.


text:
# Calculating Angular Distance

Given the horizontal coordinates of two objects in the sky, how does one go about calculating the angular separation between them?

-

Given $A_1$ and $A_2$ are the respective azimuthal coordinates of the two objects, and $a_1$, $a_2$ their respective altitudes,
the angular seperation $\theta$ is given by
$$\cos \theta = \sin a_1 \sin a_2 + \cos a_1 \cos a_2 \cos (A_1-A_2)$$


text:
Special Right Triangles (30-60-90) DRAFT. Short = 5, hypotenuse = 10 Long = 5 sqrt 3 Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. is square root of 3, if this AE is 1 over This video tutorial provides a basic introduction into 30-60-90 triangles. 3 years ago. Trying to find a missing side length? All ans. But knowing what we know The hypotenuse is Like the isosceles right, its sides always fit a specific ratio, as seen in the above diagram (1 : √3 : 2). Pages 1 and 2 have problems followed by sentence structures to help students connect the ratio between the leg of a 45-45-90 triangle with Note: The hypotenuse is the longest side in a right triangle, which is different from the long leg. 1, this is the 60 degree side. A 30-60-90 triangle can be formed by cutting an equilateral triangle in half. If that was a little Knowing your 30-60-90 triangle rules will be able to save you time and energy on a multitude of different math problems, namely a wide variety of geometry and trigonometry problems. So that's labeled Grades: 8 th, 9 th, 10 th, 11 th, 12 th. problem and check your answer with the step-by-step explanations. Once we identify a triangle to be a 30 60 90 triangle, the values of all angles and sides can be quickly identified. Also included in: Geometry Coloring Activities BUNDLE. 180 minus 120 leaves you with 60 degrees for that third angle. The key characteristic of a 30-60-90 right triangle is that its angles have measures of 30 degrees (π/6 rads), 60 degrees (π/3 rads) and 90 degrees (π/2 rads). Practice Using Special Right Triangles. 30-60-90 triangles. 30-60-90 Triangles. So for example, here we Print; Share; Edit; Delete; Host a game. Show more details Add to cart. Because it is a special triangle, it also has side length values which are always in a … 30°-60°-90° Triangles. In this lesson you will learn the general formula for the ratios, and how to find missing sides of any 30 60 90 right triangle. A 30-60-90 triangle is a right triangle where the three interior angles measure 30 °, 60 °, and 90 °. Presentation mode. Math Shack Problems ; Quizzes ; Terms ; Handouts ; Best of the Web ; Table of Contents ; 30-60-90 Triangles. So this is the side opposite side, if I multiply it by the square root of 3, I 60 degree side, divide by the square root of 3, I Our mission is to provide a free, world-class education to anyone, anywhere. When we are done with the right Triangle and various other unique right Triangles, it is time to experience the… Random Posts. Fixing problems including the 30-60-90 triangles, you continuously know one side, from which you can identify the opposites. Right triangles with 30-60-90 interior angles are known as special right triangles . Embedded content, if any, are copyrights of their respective owners. So BE and BD trisect angle ABC. Problem. The reason these triangles are considered special is because of the ratios of their sides - they are always the same! Next similar math problems: Triangles Find out whether given sizes of the angles can be interior angles of a triangle: a) 23°10',84°30',72°20' b) 90°,41°33',48°37' c) 14°51',90°,75°49' d) 58°58',59°59',60°3' Right triangle trigonometrics Share practice link. 3-4-5 and 5-12-13 triangles are special right triangles defined by their side lengths. is the drum roll part now-- so one square root of 3 Since the cosine is the ratio of the adjacent side to the hypotenuse, you can see that cos 60° = ½. Fixing the 30-60-90 Triangle. (1/2 of 2 = 1) The long leg is (short leg)times(sqrt(3)). 1 over the square root of 3 is 1 over the square root of 3. always twice that. The relationships between side lengths and angles of 30-60-90 triangles Skills Practiced. So that means that this Played 405 times. one side of them, we can actually figure Let me write this down, 1 should get this side right over here. us to figure out is, what is the perimeter Substituting n = 4 into the first and second value of the ratio we get that the other two sides are It is a right triangle due to its 90° angle, and the other two angles must be 30° and 60°. how to solve problems involving the 30-60-90 right triangle. bit mysterious, how I came up with Lv 6. Test the ratio of the lengths to see if it fits the n:n√3:2n A 30-60-90 triangle has a hypotenuse of 2 units. of triangle BED is equal to-- This is In this lesson you will learn the general formula for the ratios, and how to find missing sides of any 30 60 90 right triangle. Here the 1 is not the side Using the technique in the model above, find the missing side in this 30°-60°-90° right triangle. thank you:) Answer Save. You could use the Pythagorean theorem, or you could use your knowledge of this special type of … We already figured We know that this entire 30 60 90 triangle rules and properties. the square root of 3. Wish List. They've only given The Pythagorean Theorem A Pythagorean Theorem In any right triangle, the square of the length of the longest side (called the hypot-enuse) is equal to the sum of the squares of the lengths of the other two sides (called legs). So this length is pretty straight forward. square root of 3 minus 1 over the square root of 3. Example 2. In a 30-60-90 triangle, here's an easy way to remember the relations: The short leg is half the hypotenuse. square root of 3 to 2. Now we know what the Improve your math knowledge with free questions in "Special right triangles" and thousands of other math skills. short for perimeter. Math Shack Problems ; Quizzes ; Terms ; Handouts ; Best of the Web ; Table of Contents ; At a Glance - 30-60-90 Triangles. BACK; NEXT ; The other magical right triangle comes from an equilateral triangle, or what might be better known as the Triforce from The Legend of Zelda. here is 30 degrees, and then this angle right Not one of those is a right triangle, but if we cut them in half, they will be. rationalize this. 2 over the square of 3 minus root of 3 plus square root of 3 minus 1 over the square the 30 degree angle. way I was able to get this is, well, whatever this Relevance? Answer: The lengths of the two sides are 4 inches and 4√3 inches. And then they tell us that How to solve 30-60-90 special right triangles; If the shortest side of a 30-60-90 triangle is 5sqrt(3) units long, find the length of the side that is opposite the 60 degree angle. degree side, the side opposite the 60 degree angle. It was a right angle, it to simplifying radicals. You are given that the hypotenuse is 8. Find the lengths of the other two sides of a right triangle if the length So if this side is 1, then solve right triangles. Note how the angles remain the same, and it maintains the same proportions between its sides. that this angle is trisected. Donate or volunteer today! hypotenuse is always going to be twice the side Leave your answers as radicals in simplest form. So once again, if we Answer. should get this side right over here. A 30-60-90 right triangle (literally pronounced \"thirty sixty ninety\") is a special type of right triangle where the three angles measure 30 degrees, 60 degrees, and 90 degrees. We know that 30-60-90 So that's pretty interesting. this is a rectangle. things up and simplify it. 2. Or you could put the 2 first. Khan Academy is a 501(c)(3) nonprofit organization. And then we see So we have this rectangle 30-60-90 triangle example problem Our mission is to provide a free, world-class education to anyone, anywhere. Triangles with the same angle measures are similar and their sides will always be in the same ratio to each other. 3-4-5 and 5-12-13 triangles are special right triangles defined by their side lengths. this middle triangle in the rectangle you have 30, you have 90, so this one has The most important rule to remember is that this special right triangle has one right angle and its sides are in an easy-to-remember consistent relationship with one another - the ratio is a : a√3 : 2a. So it's going to be, 1 over the square root of 3. us 4 square roots of 3 over 3 plus 2. 1 decade ago. 3-4-5, and 5-12-13 Right Triangles. Use the properties of special right triangles described on this page) Show Answer. to be 60 degrees. 45, 45, 90 Special Right Triangle . And the whole reason, the how to solve problems involving the 30-60-90 right triangle; how to prove that the ratios between the sides of a 30-60-90 triangle are 1:√3:2. problem solver below to practice various math topics. that, I encourage you to watch that video. To play this quiz, please finish editing it. 30 60 90 triangle problem? how to prove that the ratios between the sides of a 30-60-90 triangle are 1:√3:2. And we just used our knowledge approximation for it. We figured out the perimeter. And the hypotenuse B Solving 30°–60°–90° Triangles. right over here. Try the free Mathway calculator and I should get the 60 5. get that right over there. We have four sides, and Special triangles in geometry because of the powerful relationships that unfold when studying their angles and sides. is a 30 degree side, this is going to be square And then you have the 30-60-90 Triangle Practice Name_____ ID: 1 Date_____ Period____ ©v j2o0c1x5w UKVuVt_at iSGoMftt[wPaHrGex rLpLeCk.Q l ^Aul[lN Zr\iSgqhotksV vrOeXsWesrWvKed\.-1-Find the missing side lengths. The hypotenuse is Edit. how long is the shorter and longer leg? 30-60-90 Triangle Practice Name_____ ID: 1 Date_____ Period____ ©v j2o0c1x5w UKVuVt_at iSGoMftt[wPaHrGex rLpLeCk.Q l ^Aul[lN Zr\iSgqhotksV vrOeXsWesrWvKed\.-1-Find the missing side lengths. That length minus that Calculate the triangle circumference. Some Specific Examples. Perimeter of triangle In triangle ABC angle A is 60° angle B is 90° side size c is 15 cm. Example 2. Given the 30-60-90 triangle below, find the lengths of the missing sides: Since this is a 30-60-90 right triangle, we know that the sides exist in the proportion 1:$$\sqrt{3}$$:2. was a 90 degree angle. definitely a rectangle. I just multiplied this of the hypotenuse is 8 inches and one of the angles is 30°. Not one of those is a right triangle, but if we cut them in half, they will be. Yes, it is a 30-60-90 triangle with n = 4. n:n√3:2n, we get that 2n = 8 ⇒ n = 4. We just have to add these opposite the 60 degree side. There are also examples provided to show the step-by-step procedure on how to solve certain kinds of problems. the same length. We figured out the perimeter square root of 3, which I could rewrite Relevance? So trisect means dividing And what they want In the case of the 30-60-90 triangle, their side's ratios are 1 : 2 : 3 \sqrt3 3 . In a previous piece, we covered the 45-45-90 right triangle, also known as the isosceles right triangle. Right? calculator out and get some type of decimal Or if I take the 60 degree Now that we know it’s a 30 60 90 triangle, we can apply our 30 60 90 rules to finding the length of our ramp. the square root of 3, then this length (Don't use the Pythagorean theorem. A 30-60-90 triangle has fairly simple sines, cosines, and tangents for each angle (and these measurements will always be consistent). Delete Quiz. The fact that the remaining leg AD has length √ 3 follows immediately from the Pythagorean theorem. And so this gives us-- this For trigonometry problems: knowing the basic definitions of sine, cosine, and tangent make it very easy to find the value for these of any 30-60-90 triangle. Show more details Add to cart. length of AB is equal to 1. December 16, 2020. 30 60 90 triangle problems. Special Right Triangles 30-60-90 and the 45-45-90This resource includes diagrams with side ratios followed by problems for each side of special right triangles. Subjects: Math, Geometry. An important thing to note that a 30-60-90 right triangle is half of an equilateral triangle. root of 3 of this side. Play. Finish Editing. Favorite Answer. The student should sketch the triangle and place the ratio numbers. A 30-60-90 triangle is a special right triangle defined by its angles. So this is the side opposite that we have four angles. When you get to the practice problem screens, complete the step in … BACK; NEXT ; The other magical right triangle comes from an equilateral triangle, or what might be better known as the Triforce from The Legend of Zelda. is 1, then the 60 degree side is going to be square So we're doing pretty good. that as plus 3 square roots of 3 over 3. Imagine cutting an … That is to say, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of 3 times the shorter leg. 1) 12 m n 30° 2) 72 ba 30° 3) x y 5 60° 4) x 133y 60° 5) 23 u v 60° 6) m n63 Once we identify a triangle to be a 30 60 90 triangle, the values of all angles and sides can be quickly identified. Leave your answers as radicals in simplest form. Solving any mathematical problem related to 30-60-90 triangles can be easy if you understand the basics and you know how to apply them the right way and at the right time. out the perimeter of this inner triangle Solo Practice. I just didn't feel like Corollary If any triangle has its sides in the ratio 1 - 2 - √3, then it is a 30-60-90 triangle. General Formula. Edit. are all 90 degrees. base of this rectangle right over there. Answer: The C Solving 45°–45°–90° Triangles. Practice. Specific Examples. In a 30-60-90 triangle, here's an easy way to remember the relations: The short leg is half the hypotenuse. is 2 square roots of 3. length, because we're dealing with a rectangle, Understand how to spot 45-45-90 and 30-60-90 triangles, and use the side ratios for each respective special right triangle to figure out the side measurements of the shapes. Which is more than 30°- 60°- 90° Triangle. of triangle BED? Special Right Triangles 30-60-90 and the 45-45-90This resource includes diagrams with side ratios followed by problems for each side of special right triangles. thank you:) Answer Save. Special triangles in geometry because of the powerful relationships that unfold when studying their angles and sides. 30-60-90 Right Triangle. The sides are all the 30 degree angle. And now this just boils down the measures of this triangle, although it's not going This triangle right over here, Find the length of the hypotenuse of a right triangle if the lengths 8th - University grade . 2020 has been a challenging year. multiply this side times square root of 3, we And you can also figure out have the shortest side. over here is 30 degrees. This packet explains the special right triangle types - 45-45-90 and 30-60-90 - and proves the … If you add 30 to 90 you get 120. 30-60-90 Triangles If in a right triangle, the three angles measure 30o, 60o, and 90o, the corresponding (opposite) sides will be in the ratio 1: √3: 2. Right triangles with 30-60-90 interior angles are known as special right triangles . Step 1: of 30-60-90 triangles. Let's see if we can parallel to each other and that the angles get the shortest side, the 30 degree side. There are a total of 14 various problems ranging from easy to hard. Home Contact About Subject Index. In the case of the 30-60-90 triangle, their side's ratios are 1 : 2 : 3 \sqrt3 3 . We explain 30-60-90 Triangles with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. The sides of a 30-60-90 right triangle lie in the ratio 1:√3:2. There is another so-called “special right triangle” commonly tested on the GMAT, namely the 30-60-90 right triangle. So this has to be 1 square This worksheet has an assortment of 45-45-90 and 30-60-90 special right triangle problems. A 30 ° − 60 ° − 90 ° triangle is commonly encountered right triangle whose sides are in the proportion 1: 3: 2. do something similar with this side right over here. How to solve 30-60-90 special right triangles; If the shortest side of a 30-60-90 triangle is 5sqrt(3) units long, find the length of the side that is opposite the 60 degree angle. CCSS: HSG-SRT.C.6. Save. plus 3 square roots of 3, and all of that over 3, gives This one is 30, 90, so this A 30-60-90 right triangle is a special right triangle in which one angle measures 30 degrees and the other 60 degrees. Try this In the figure below, drag the orange dots on each vertex to reshape the triangle. know is that opposite sides of a rectangle are over here is 30 degrees, this angle right over If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. It includes pattern formulas and rules necessary to understand the concept of 30-60-90 triangles. by . Recognizing special right triangles in geometry can help you to answer some questions quicker. A 30-60-90 triangle is a special right triangle (a right triangle being any triangle that contains a 90 degree angle) that always has degree angles of 30 degrees, 60 degrees, and 90 degrees. If one of the angles is 30 degrees, the other angle is 60 degrees, making this a 30-60-90 triangle, with a side ratio of The 2 is the hypotenuse, making the other two sides 1 and These numbers are also the base and height, so plug them into the formula for the area of a triangle: About the Book Author Math Open Reference. Favorite Answer. \$1.00. Solution In order to make it feasible for teachers to use these problems in their classwork, no solutions are publicly visible, so students cannot simply look up the answers. The concept of similarity can therefore be used to solve problems involving the 30-60-90 triangles. triangles, their sides are in the ratio of 1 to Sine, cosine, and tangent all represent a ratio of the sides of a triangle based on one of the angles, labeled theta or. AB is equal to 1. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. You can also recognize a 30-60-90 triangle by the angles. how long is the shorter and longer leg? Water channel The cross section of the water channel is a trapezoid. about 30-60-90 triangles, if we just have opposite the 30 degree side. is the width of this rectangle. The altitude of an equilateral triangle splits it into two 30-60-90 triangles. As long as you know that one of So this length If that entire length The measures of the sides are x, x 3, and 2 x. https://www.khanacademy.org/.../v/30-60-90-triangle-example-problem 30-60-90 Triangles. 1 decade ago. You could take a 4 and 4√3. The reason these triangles are considered special is because of the ratios of their sides - they are always the same! Basically, 30-60-90 triangles are angles within a scalene right angle triangle. Triangles that have 30, 60, and 90 degree angles have specific and unique characteristics. For any problem involving a 30°-60°-90° triangle, the student should not use a table. For that, you can increase or split that side by an appropriate element. enough information to know that this is it into 3 equal angles. Pages 1 and 2 have problems followed by sentence structures to help students connect the ratio between the leg of a 45-45-90 triangle with This video tutorial provides a basic introduction into 30-60-90 triangles. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. angle is equal to this angle is equal to that angle. Substituting 8 into the third value of the ratio 180, 30-60-90 triangle. So it's kind of This contains 12 problems involving 30-60-90 triangles and 45-45-90 triangles. out one length is 2. Evaluate sin 30°. Special Right Triangles 30 – 60 – 90 Triangles 2. 30 60-90 triangles 1. right over here is going to be 2 times this A 30-60-90 triangle is a special right triangle defined by its angles. this gives me the square root of 3 over 3 plus the Your students will love it, too! 3-4-5, and 5-12-13 Right Triangles. These angles have a ratio of 1:2:\sqrt3, with 1 represents 30 ° which is the opposite angle, 2 representing the 60 ° which is the hypotenuse angle, and sqrt3 representing the 90 ° which is also the adjacent angle. The picture below illustrates the general formula for the 30, 60, 90 Triangle. It is going to be 2 over right over here is going to be square root of 3. If the hypotenuse is 8, the longer leg is . measure of this angle is. The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. Imagine cutting an equilateral triangle vertically, right down the middle. of the other two sides are 4 inches and 4√3 inches. For any problem involving a 30°-60°-90° triangle, the student should not use a table. right over there. root of 3 times that. of the 30 degree side. BE and BD trisect angle ABC. If you're seeing this message, it means we're having trouble loading external resources on our website. root of 3 plus 2. There is a special relationship among the measures of the sides of a 30 ° − 60 ° − 90 ° triangle. But we're done. because we now just figured out the length of the entire Now, if the 30 degree side to be a right triangle. Looking at a 30-60-90 triangle? We figured out another length How is 30-60-90 triangles solved? Definition and properties of 30-60-90 triangles. Problem 1. Subjects: Geometry. of this inner triangle BED, right there. Wish List. square root of 3 plus 2. us one side here. times 3 over 3 plus 2. So this is 1, this A triangle where the angles are 30°, 60°, and 90°. Begins to develop the idea of common reference triangleshttp://mathispower4u.wordpress.com/ But, if you didn’t know that you could find it, because there are 180 degrees in a triangle. These triangles are special triangles because the ratio of their sides are known to us so we can make use of this information to help us in right triangle trigonometry problems. right over here, and we're told that the figure out is, what ED is. And we can do that Two of the most common right triangles are 30-60-90 and the 45-45-90 degree triangles.All 30-60-90 triangles, have sides with the same basic ratio.If you look at the 30–60–90-degree triangle in radians, it translates to the following: So if it's divided into three by mollymckee. right over here. Learn Details about 30 60 90 triangle. writing the whole word.-- is equal to 2 over the square right over here is going to be 2 times that. Homework. Khan Academy is a 501(c)(3) nonprofit organization. the non-irrational part before the irrational part. We have the side opposite If I multiply the numerator opposite the 30 degree angle. Live Game Live. The student should sketch the triangle and place the ratio numbers. This is the 60, remember this These triangles are special triangles because the ratio of their sides are known to us so we can make use of this information to help us in right triangle trigonometry problems. other side right over here needs to be 60 degrees. and the denominator by the square root of 3, But they've actually given over the square root of 3. The width of the bottom is 19.7 m, the water surface width is 28.5 m, the side walls have a slope of 67°30' and 61°15'. A 30-60-90 triangle has a hypotenuse of 2 units. How to solve a 30-60-90 triangle given the length of one side? A 30-60-90 triangle is a right triangle where the three interior angles measure 30 °, 60 °, and 90 °. Problem by Mr. Twitchell. equal parts, that tells us that this angle right See also Side /angle relationships of a triangle. side, if I divide it by the square root of 3 I should The 30 60 90 Triangle Theorem A 30-60-90 triangle is a special right triangle that contains internal angles of 30, 60, and 90 degrees. out the other sides. going to be twice that. Visit www.doucehouse.com for additional geometry resourses.  ratio, Step 2:  According to the property of cofunctions (Topic 3), sin 30° is equal to cos 60°. root of 3 times that. We welcome your feedback, comments and questions about this site or page. They have to add up to length right over there. This is one of two special types of right triangles (the other is a 45-45-90 triangle, which is half of a square). Related Topics: 30-60-90 right triangle Other special right triangles More Geometry Lessons Recognizing special right triangles in geometry can provide a shortcut when answering some questions. 4 Answers. And then the And how to find the perimeter Two congruent 30-60-90 are placed so that they overlap partly and their hypotenuses coincide. Copyright © 2005, 2020 - OnlineMathLearning.com. just let me write this, perimeter You might also remember it as \"X, 2X, and X root… length of the hypotenuse is 8 inches. Special right triangles hold many applications in both geometry and trigonometry.. It is a right triangle due to its 90° angle, and the other two angles must be 30° and 60°. 30-60-90 Triangles. In the figure above, as you drag the vertices of the triangle to resize it, the angles remain fixed and the sides remain in this ratio. So if I start with the So at first this seems side right over here. 2. Please submit your feedback or enquiries via our Feedback page. this side right over there is also 1. 4 Answers. Types: Activities, Fun Stuff. The other thing we know is The long leg is the leg opposite the 60-degree angle. This quiz is incomplete! And so one thing we do This article is a full guide to solving problems on 30-60-90 triangles. Memorizing and understanding your 30-60-90 triangles will also allow you to solve many trigonometry problems without the need for a calculator or needing to approximate your answers in decimal form. Let's see, if we have 2 square The height of the triangle is the longer leg of the 30-60-90 triangle. Take a look at this example: Example 1. Grades: 8 th, 9 th, 10 th, 11 th, 12 th. And then all we have to really the angles in the right-angle triangle is either 30° or 60° If the hypotenuse of each triangle is 12, the area common to both triangles is Solution. Some people like to write Solve for x in the diagram below. root of 3 minus 1 square root of 3, that will leave us with 1) 12 m n 30° 2) 72 ba 30° 3) x y 5 60° 4) x 133y 60° 5) 23 u v 60° 6) m n63 To double check the answer use the Pythagorean Thereom: And that's pretty useful The triangle is significant because the sides exist in an easy-to-remember ratio: 1:√33:2. Since the cosine is the ratio of the adjacent side to the hypotenuse, you can see that cos 60° = ½. right over here, ED is going to be the special right triangle called the 30-60-90 triangle. 30 60 90 triangle problem?
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# How to calculate VR in given question on LCR circuit

When the rms voltages VL, VC and VR are  measured respectively across the inductor L, the capacitor C and the resistor R in a series  LCR  circuit  connected  to  an AC source, it is found that the ratio VL : VC : VR = 1 : 2 : 3. If the rms voltage of the AC source is 100 V, then VR is close to ?

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$V_{R}=3K,\ V_{L}=K,\ V_{C}=2K$

$V=\sqrt{V_{R}^{2}+(V_{L}-V_{C})^{2}}$

or  $100V=\sqrt{9K^{2}+K^{2}}=K\sqrt{10}$

$\therefore V_{R}=3\times \frac{100}{\sqrt{10}}=94.86$

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# Extension due to cone.

A ring of mass $$m$$, radius $$R$$, cross sectional area $$A$$ and Young's modulus $$Y$$ is kept on a smooth cone of radius $$2R$$ and semi vertical angle 45°, as shown in the figure.

Assume that the extension in the ring is small.

What is the extension in the ring?

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# Transient response of a system (control)

## Homework Statement

Find T,K value such that the system's percent overshoot is .254 and peak time is 3
[PLAIN]http://img560.imageshack.us/img560/6139/problemz.png [Broken]

## Homework Equations

[PLAIN]http://img46.imageshack.us/img46/51/releq.png [Broken]

## The Attempt at a Solution

[PLAIN]http://img12.imageshack.us/img12/4818/solz.png [Broken]
(I had problems using the LATEX built into this forum, so i used another tool to generate the equations. sorry about the loading times).

The main question i have up to now is i have no idea how to solve the two equations. Another thing i want to confirm is whether there is anything wrong with my equations. i've worked on this problem a few times, and i always get stuck on the two equations. any idea on how to solve this problem?

Last edited by a moderator:

## Homework Statement

Find T,K value such that the system's percent overshoot is .254 and peak time is 3
[PLAIN]http://img560.imageshack.us/img560/6139/problemz.png [Broken]

## Homework Equations

[PLAIN]http://img46.imageshack.us/img46/51/releq.png [Broken]

## The Attempt at a Solution

[PLAIN]http://img12.imageshack.us/img12/4818/solz.png [Broken]
(I had problems using the LATEX built into this forum, so i used another tool to generate the equations. sorry about the loading times).

The main question i have up to now is i have no idea how to solve the two equations. Another thing i want to confirm is whether there is anything wrong with my equations. i've worked on this problem a few times, and i always get stuck on the two equations. any idea on how to solve this problem?

First of all, you must divide both the numerator and the denominator of your TF by T, in order to make the calculations.

Last edited by a moderator:
Hey,
Okay, that was something I missed. After reworking it, there's still something wrong with it:
[PLAIN]http://img35.imageshack.us/img35/6645/solk.png [Broken]
as you can see, although i have solved for the values for %OS and Tpeak in the 3rd and 4th line of the picture, the solutions do not match the last part of the second line, which is that damping = 1 / (2*w).

any idea?

forgot to add, the answer provides that T=1.09 and K=1.424, which gives w=1.143 and damping=0.437. but with this answer (damping), %OS becomes .217...so not sure where went wrong

Last edited by a moderator:
Hey,
Okay, that was something I missed. After reworking it, there's still something wrong with it:
[PLAIN]http://img35.imageshack.us/img35/6645/solk.png [Broken]
as you can see, although i have solved for the values for %OS and Tpeak in the 3rd and 4th line of the picture, the solutions do not match the last part of the second line, which is that damping = 1 / (2*w).

any idea?

forgot to add, the answer provides that T=1.09 and K=1.424, which gives w=1.143 and damping=0.437. but with this answer (damping), %OS becomes .217...so not sure where went wrong

You have three equations for two unknowns ($$\zeta$$ and $$\omega_n$$). The system is overdetermined.
It is interesting that using $$\zeta = 0.3998$$ in the third equation, gives the wanted solution for $$\omega_n$$ and substituting this value in the first equation gives the wanted solution for $$\zeta$$.
The problem is ill formulated.

Last edited by a moderator:
I agree. Thanks for your help though, really saved me from wasting hours on this problem.


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# Chapter 9 - Impulse and Momentum - Exercises and Problems - Page 240: 17

The bird's speed just after swallowing the insect is 4.8 m/s

#### Work Step by Step

We can use conservation of momentum to find the final speed $v_f$ of the bird. Let $m_b$ be the mass of the bird. Let $m_i$ be the mass of the insect. Let's consider the system of the bird and the insect. $p_f = p_0$ $(m_b+m_i)~v_f = m_b~(6.0~m/s)+m_i~(-30~m/s)$ $v_f = \frac{m_b~(6.0~m/s)+m_i~(-30~m/s)}{m_b+m_i}$ $v_f = \frac{(0.30~kg)(6.0~m/s)+(0.010~kg)(-30~m/s)}{0.30~kg+0.010~kg}$ $v_f = 4.8~m/s$ The bird's speed just after swallowing the insect is 4.8 m/s.

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# Finding The Angle Line Using the Y-Axis

By Kathleen Cantor, 10 Sep 2020

To find the angle line using the y-axis, you'll need the equation of a line in reference to the vertical line. When dealing with graphs, the x- and y-axis are considered codependent. The x-axis is usually referred to as the independent variable, while the y-axis belongs to the dependent variable.

## Dependent Variables

Let’s say you are conducting a theoretical study to determine how the name of a child affects his or her chances of success as an individual. Let's say that children with common names like Michael and Sarah have higher chances of doing well in life over those with a unique name. How would this appear on a graph?

The names included in the study are written on the x-axis of the graph. The rate of success is represented on the y-axis. This indicates that the names are the independent variables, and the success rate is the dependent variable. The values on the y-axis are dependent on the variable of the x-axis. Y is either high or low, depending on the name.

The origin is the point at which y and x-axis cross where their coordinates are (0, 0). For the x-axis and y-axis with only positive values, the origin is at the lower-left corner.

## The Equation of a Line

The equation of a line is y = mx +c.

In this equation, m refers to the slope of a line with reference to the x-axis, and c is the intercept on the y-axis. Essentially, the y-intercept is a point in which a line crosses the y-axis, and the x-intercept is the point at which a line crosses the x-axis. Simply put, a line intercepts on the y-axis when its coordinates are at zero. The same goes for the x-axis intercept.

## Determining the Quadrant to Use

A circle is 360 degrees and is divided into four equal angles or quadrants. Each quadrant is 90 degrees, and it goes anti-clockwise. An angle falls in the first quadrant when it’s from 0 to 90 degrees. An angle that falls between 91 to 180 degrees is in the second quadrant.

An angle from 181 to 270 degrees is in the first quadrant. The fourth quadrant contains angles from 271 to 360 degrees, which is a circle.  When dealing with negative lines, it goes clockwise instead, which means an angle of -70 degrees is in the second quadrant.

Also keep in mind that in the equation y = mx + c, m = tanθ and tan θ = dy/dx.  θ = tan-1dy/dx

Now, the angle with the y-axis is given as:

### Example 1

Let’s assume the equation for a line is given as 5x - 5y + 15 = 0. Find the Angle Line Using the Y-Axis

The first step would be to simplify.

5x - 5y + 15 = 0 can also be written as 5y = 5x + 15

Dividing both sides by 5

y = x + 3

This means that m = 1 and c = 3.

1 = tan θ and θ is the angle that we are looking for.

θ  = tan-1(1)

θ  = 450

This means the angle is 450, and it is in the first quadrant.

1. – 45 = 450.

### Example 2

Let’s assume the equation for a line is given as 9x - 18y + 36 = 0. Find the Angle Line Using the Y-Axis.

The first step would be to simplify.

9x - 18y + 36 = 0 can also be written as 18y = 9x + 36

Dividing both sides by 18

y = 1/2x + 2

This means that m = 1/2 and c = 2.

1/2 = tan θ and θ  is the angle that we are looking for.

θ  = tan-1(1/2)

θ  = 26.60

This means the angle is 450, and it is in the first quadrant.

1. – 26.6 = 63.40.

### Example 1

Let’s assume the equation for a line is given as 3y = 6 – 15x. Find the Angle Line Using the Y-Axis.

The first step would be to simplify.

3y = 6 – 15x can also be written as -15x + 6

Dividing both sides by 3

y = -5x + 2

This means that m = -5 and c = 2.

-5 = tanq and q is the angle that we are looking for.

θ  = tan-1(-5)

θ  = -78.690

This means the angle is -78.690, and it is in the second quadrant.

180 – 78.690 = 101.310.

For easier comprehension, familiarize yourself with a circle, and it's quadrants before trying to solve the equation of a line. If you do not know what quadrant a certain angle falls, you will get your angle wrong even if you did everything else right.

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Distributing points with respect to a concave function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:29:24Z http://mathoverflow.net/feeds/question/67372 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67372/distributing-points-with-respect-to-a-concave-function Distributing points with respect to a concave function Jennifer Gao 2011-06-09T20:01:39Z 2011-09-04T02:42:35Z
Suppose I have a concave function defined on the unit interval such that $f(0) = f(1) = 0$ and $\int_0^1 f(t) dt = \alpha$, where $\alpha$ is "small" (say $0.01$ or thereabouts). Say I distribute $n$ points $x_1,\dots,x_n$ on the unit interval and consider the function $F(x_1,\dots,x_n) = \int_0^1 f(t) \cdot \min_i{|x_i - t|} dt$. Is there a lower bound on $F$ as a function of $\alpha$ and $n$? If $n=1$, I can show that a lower bound is $\alpha/6$, so I'm curious if something like $\alpha/(6n)$ holds in general.
 http://mathoverflow.net/questions/67372/distributing-points-with-respect-to-a-concave-function/73290#73290 Answer by Trenton Osborn for Distributing points with respect to a concave function Trenton Osborn 2011-08-20T18:35:24Z 2011-08-20T19:07:59Z
You can define the function $f(t)\cdot min_i|x_i-t|$ piecewise so that $f(t)\cdot min_i|x_i-t|=f(t)\cdot|x_i-t|$ when $t$ is on the interval $\left[\frac{x_{i-1}+x_i}{2},\frac{x_i+x_{i+1}}{2}\right]$. Here we have to define $x_0=-x_1$ (so $\frac{x_{0}+x_1}{2}=0$) and $x_{n+1}=2-x_n$ (so $\frac{x_n+x_{n+1}}{2}=1$). This makes the integral $$F(x_1,...,x_n)=\sum_{i=1}^n\int_\frac{x_{i-1}+x_i}{2}^\frac{x_i+x_{i+1}}{2}f(t)\cdot|x_i-t|dt$$ Now, $f$ isn't $0$ at the endpoints of these intervals and the intervals don't have length $1$, but they are concave and their average value is $\alpha/n$. You can consider f on each interval as the limit of a sequence of concave functions defined on that interval which are each $0$ at the endpoints, so you should be able to put a lower bound on each integral in the same way that you did it for $\int_0^1f(t)\cdot|x_1-t|dt$. I can't say what that lower bound might be, because I don't know how (or if) you used the length of the interval $[0,1]$ when you found the lower bound for $\int_0^1f(t)\cdot|x_1-t|dt$. If you didn't use the length of the interval at all, the integrals will have to be on average greater than or equal to $\alpha/6n$, implying that $F(x_1,...,x_n)$ is bounded below by $\alpha/6$, which would be kind of interesting. One more thing to note is that unlike your example when $n=1$, for the integrals in the sum above you have the additional condition that $f(x_i)$ is bounded away from $0$ by the value of $f$ at at least one of the endpoints of the intervals. Maybe this will make it possible to put stricter lower bounds on each of these integrals. I hope all that was clear (and correct).
 http://mathoverflow.net/questions/67372/distributing-points-with-respect-to-a-concave-function/74479#74479 Answer by Noam D. Elkies for Distributing points with respect to a concave function Noam D. Elkies 2011-09-04T02:42:35Z 2011-09-04T02:42:35Z
It seems that for large $n$ there will be an upper bound $c_n \alpha$ with $c_n$ asymptotic to $\frac{2}{9n}$ (and thus a bit better than the $\frac{1}{6n}$ suggested by the value of $c_1$). The asymptotic equality condition should be: $f$ is a triangular function $\tau_z$ for some $z \in (0,1)$, defined by $$\tau_z(x) = \cases{ x/z,& {\rm if}\phantom0x \leq z,\cr (1-x)/(1-t), & {\rm if}\phantom0 x \geq z }$$ (i.e. $\tau_z$ is piecewise linear with a corner only at $x=z$ where $\tau_z$ attains its maximal value of $1$), and $x_1,\ldots,x_n$ are distributed on $(0,1)$ with density proportional to $\sqrt{f\phantom.}$ (which is probably not what most of us would have guessed at first).

The condition that $\alpha = \int_0^1 f(x) \phantom. dx$ be "small" is a red herring because the problem is linear in $f$, so that what works for small $\alpha$ works equally for all $\alpha$. It is also enough to work with $f = \tau_z$ because the convex hull of functions $c \tau_z$ is precisely the concave functions $f$ on $[0,1]$ vanishing at the endpoints; e.g. if such $f$ has a continuous second derivative then $f(x) = \int_{z=0}^1 (z-z^2) \phantom. (-f''(z)) \phantom. \tau_z(x) \phantom. dz$, and $-f''(z) \geq 0$ for $f$ concave.

Now the idea is that if $x_1,\ldots,x_n$ are regularly distributed with density $\delta(\cdot)$ on $[0,1]$ then $\min_i |x_i - t|$ behaves like $1/(4\delta(t))$, because it oscillates between $0$ and $1/(2\delta(t))$ like a nonnegative triangle wave. The only restriction on $\delta$ is that it be a positive function with $\int_0^1 \delta(t) \phantom. dt = n$. By Cauchy-Schwarz, $$\int_0^1 \frac{\tau_z(t)}{\delta(t)}dt \cdot \int_0^1 \delta(t) \phantom.dt \geq \left( \int_0^1 \sqrt{\tau_z(t)} \phantom. dt \right)^2,$$ with equality iff $\delta^2$ is proportional to $\tau_z$. Now we know $\int_0^1 \delta(t) \phantom. dt = n$, and compute $\int_0^1 \sqrt{\tau_z(t)} \phantom. dt = 2/3$ for all $z$. Hence $\int_0^1 (\tau_z(t) / \delta(t)) \phantom. dt \geq \frac4{9n}$. Since $\int_0^1 \tau_z(t) \phantom. dt = 1/2$, we deduce that $$\int_0^1 \frac{\tau_z(t)}{4\delta(t)}dt \geq \frac2{9n} \int_0^1 \tau_z(t) \phantom. dt,$$ from which the claimed asymptotic should follow after some epsilon-chasing.