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chemical equation writer for word

Sorry for bothering you and thanks again for your great video. You can also assign a keyboard shortcut to the InsertEquation command.] A chemical reaction occurs when starting substances react to produce new substances. Enter an equation of a chemical reaction and click 'Submit' (for example: 3Mn2++2MnO4-+2H2O = 5mno2+ 4h^+ or cuso4*5h2o(s)-->cuso4*h2o(s)+4h2o(g)). Style UI components with CSS; Fine control over structure drawing; Compatible with dynamic content generation; No More Raster Images. Silver sulfide is commonly known as tarnish and turns the surface of silver objects dark and streaky black (see Figure below). Insert a check mark or other symbol Add your answer and earn points. Equation With Variable On Both Sides Calculator. I’m now a PhD student in Earth and Planetary Science field. The method that I am talking is called Math Autocorrect which was introduced in Ms Word. But don’t worry you have come to right place. Use uppercase for the first character in the element and lowercase for the second character. This free download adds support for entering and editing chemistry symbols, diagrams, and formulas using the standard XML based Chemical Markup Language. 0.2 Write word equations; Topic 3 - Chemical changes. By using this method you can type chemical reaction in Microsoft Word around 10 time faster than normal method of using Graphical User Interface (GUI) of Ms Word. And, if you need any help or want to learn more about the Chemistry add-in and its features, simply click the help button in the Chemistry Ribbon. The sulfur that contributes to tarnish can come from traces of sulfur in the air or from food such as eggs. Now you will have a new Chemistry tab in your Word ribbon. To balance a chemical equation, enter an equation of a chemical reaction and press the Balance button.  . Tip: To make sure that size of arrow fits text length, first type the longest text above or below arrow and press space then type another text below/above. This chemistry video tutorial explains how to write chemical equations from word descriptions. Hydrogen reacts with oxygen even below its lower flammability (4% V/V) limit in presence of catalyst like Pt and Pd. However, all the shortcut that I have covered in video is given https://www.cs.bgu.ac.il/~khitron/Equation%20Editor.pdf. We use cookies to ensure that we give you the best experience on our website. Just drag them into the view and start your work. Equation label . Matthew Guay is a veteran app reviewer and technology tip writer. You can type all types of Chemical reaction in word by going into Insert-> Equation. Spaces is an important part of Math AutoCorrect shortcut. I was wondering if the document is no longer accessible now. Regards Pickupbrain. BeCl2(aq) + 2 AgNO3(aq) ( … The above code will correctly handle chemical equations like 'C5H8(N2S)4' and '3(CaO)•2(SiO2)•4(H2O)(gel) + 3Ca(OH)2'. Now you can see the chemical form in the sidebar if you click the Chemistry Navigator button, and can choose to insert the diagram into the document. shortcut for writing subscript and superscript, shortcut for typing chemical equation in word, Python Tutorial : Use Map Function to speedup your python code, Windows automation tool Autohotkey | Automate common task with Autohotkey |, https://www.youtube.com/watch?v=O8eJ6Xi6MN8, https://www.cs.bgu.ac.il/~khitron/Equation%20Editor.pdf, Default font and font size in Microsoft Word, Concatenate Two Or More Dictionaries In Python, How to set default font in Word – Smart Word User, Automated interpolation formula for Excel: Define excel interpolate function & use it forever. When writing a chemical equation you can use two methods - using symbols to represent the chemicals or by using words. Trending Posts. Microsoft Word is a great tool for formatting text, but what if you want to insert a chemistry formula or diagram? You can also convert a chemical formula directly to it’s chemical diagram. I would like to hear from you. Some chemicals will automatically convert to the diagram in the document, while others … Is signifies direction of reaction and many more things. Comment on Uttrakhand Pravasi Registration Form – उत्तराखंड यात्रा पंजीकरण प्रपत्र – Online Form by Seema sharma Uttrakhand Pravasi Registration Form – उत्... Ms Word Tutorial on How to type chemical reaction in Word: Chemical formula, Reaction arrows and text above and below arrow, Click to share on Twitter (Opens in new window), Click to share on Facebook (Opens in new window), Click to share on LinkedIn (Opens in new window), Click to share on Reddit (Opens in new window), Click to share on Tumblr (Opens in new window), Click to share on Pinterest (Opens in new window), Click to share on Telegram (Opens in new window), Click to share on WhatsApp (Opens in new window). Choose Insert > Equation and select Insert New Equation from the bottom of the built in equation gallery. Thanks for contacting us. Chemical Equations. digvijay49 digvijay49 Answer: Hope you like this answer. The symbols and formulae of the substances (elements or compounds) are arranged to show the reactants and products of a chemical reaction. Every shape can be edited and rearranged. If your document has other alphanumeric strings in which a non-superscripted number follows a letter (e.g. At many place we need to type text above and below arrow to show catalyst, reaction conditions etc. This inserts an equation placeholder where you can type your equation. However, I could’t find the word document link in the video description as you had mentioned in video. Later, printed cookbooks became available (even guys had no excuse for not being able to cook). 0.2 Write word equations; 0.3 Write balanced chemical equations, including the use of the state symbols (s), (l), (g) and (aq) Edexcel Combined science. First step of writing equation in Ms Word is to get equation editor. Explanation: In order to write a word equation to represent a chemical reaction I must: write the name of the reactant (carbonic acid) draw an arrow to show the direction of the chemical reaction, → write the names of each of the products separated by a plus sign (+) to the right of the arrow (carbon dioxide + water) Informally called Chem4Word, this free tool is being unveiled today during the American Chemical Society’s Spring 2010 National Meeting & Exposition. You will be asked if you want to install this customization, so click Install to enable it. A chemical equation is a representation of a chemical reaction that displays the reactants and products with chemical formulas. Whether you’re a professional chemist, just taking chemistry in school, or simply curious about the makeup of Citric Acid, this add-in is an exciting way to bring chemistry to your computer. Write word equations for the reactions in this specification. Here is how it looks like (see the x2 part highlighted in orange box): Now, type 2. Here’s the ribbon in Word 2010…. Non-blocking I/O functions; Single global object; Unobtrusive startup; Customize. Free Chemical Reactions calculator - Calculate chemical reactions step-by-step. Some chemicals will automatically convert to the diagram in the document, while others simply link to it in the sidebar. All chemical reactions are represented by chemical equations. To type an equation from scratch, press Alt += on your keyboard. https://www.youtube.com/watch?v=O8eJ6Xi6MN8 WRITING CHEMICAL EQUATIONS ON THE WEB. Features:edit document, insert clipart or pictures, Equation Editor, etc. Here is how it looks like (see the x2 part highlighted in orange box): Now, type 2. If you choose to edit the chemical in 2D, you can even edit the individual atoms and change the chemical you’re diagramming. When you click Edit Labels, you may be asked which form you wish to display. Here’s the options for potassium permanganate: You can then edit the names and formulas, and add or remove any you wish. . The Chemical Equation Drawing Software includes some pre-defined molecular shapes. A round of applause for your blog article. By using this website, you agree to our Cookie Policy. Equation: Chemical equation Equation with oxidation numbers. I am in a process of making a document but may take some time. Formulae, equations and hazards. The Equation Editor is available from the Document Elements tab or by using the Insert> Equation command in the menu. WRITING CHEMICAL EQUATIONS ON THE WEB. In your Word file, to type a formula, for example H2SO4. Plans. Since we launched in 2006, our articles have been read more than 1 billion times. Left arrow Right arrow Double arrow. A chemical equation is a shorthand representation of a chemical reaction using the symbols and formulae of substance involved in the chemical reaction. Unique interactive multimedia Chemistry teaching software that tests students as they learn. Equation editor shortcut for writing subscript in Ms Word is simply _ (underscore). Example here long text in below arrow so we use the following shortcut\rightarrow\below(P = 10Mpa)\above(Ni), Combining all the shortcut into one example. IE9+, Chrome, Firefox, Safari, Edge, Brave, and Opera ; Dedicated touch interface; Works with CSS Grid and Flexbox; Write Clean Code. All Rights Reserved. If you say "forms", "makes", or "yields", it means the following substances are products. Kingsoft Writer 2012 is a word processor that is fully compatible with all Word. Yes, we should! Writing a balanced chemical equation 4 1 and balancing how do we represent reactions ch104 chapter 5 chem11pracprobkey equations the of photosynthesis solved experiment pre laboratory. Quickstart. Chemical Structure Drawing & Display for Web Applications. This reaction could be written as. Replace immutable groups in compounds to avoid ambiguity. Equation number . table cell references), you’ll need to select only the range(s) containing the text to be converted and answer ‘No’ to the macro's 'whole document' prompt. A chemical equation tells you which chemicals reacted together (the reactants) and the new chemicals that were made in the reaction (the products). If it is still possible, I would really like to have one in hand. New functions: spell-check feature, web layout, insert page break by single-click, etc. Shortcut for all the reaction arrows like equilibrium arrow, reversible arrow etc are listed in following table. Microsoft Word Equation Editor Tutorial Author: Khitron Igal et al Subject: A simple 4-pages tutorial for MS Word equations shortcuts Keywords: Equation editor, equation, Word, Microsoft Word, shortcuts, tutorial, manual, 2010 Created Date: 1/14/2014 2:24:53 PM To write text above arrow use \above() and to write text below arrow use \below(). Align left Center A word equation shows the reactant reacting to produce the product: reactant → product . Solution for Write the chemical equation and balance the reaction for the following word equations: A solution of ammonium sulfate added to a solution of lead… There are three options here: Click the arrow next to the Equations icon for a drop-down selection of common equations. The above article may contain affiliate links, which help support How-To Geek. Enter an equation of a chemical reaction and click 'Submit' (for example: 3Mn2++2MnO4-+2H2O = 5mno2+ 4h^+ or cuso4*5h2o(s)-->cuso4*h2o(s)+4h2o(g)). The arrow in the general form of the equation above shows the direction in which the chemical reaction proceeds: that is, we start with "reactant" and we end up with "product". When silver metal is exposed to sulfur it reacts to form silver sulfide. Download the Chemistry add-in from Microsoft Education Labs (link below), and unzip the file. Word Equations. Dear Jessie Hung, Balancing Chemical Equations Examples For Class 7. Deploy Everywhere. Some of the worksheets for this concept are Word equations work, Word equations work, Balancing equations work and key 7 23 09, Chemical equation work, Word equations and balancing equations, Chemistry chemical word equations, Balancing word equations chapter 9, Work word equations name. This will open a Word document containing examples and explanations which can be helpful in mastering all the features of this add-in. Non-blocking I/O functions; Single global object; Unobtrusive startup; Customize. Here we entered H2O and converted it to Chemistry Zone: This directly converted it to the diagram directly in the document. Create Chemistry Equations and Diagrams in Word, How to Turn on Dark Mode at Sunset on Android, How to Configure Microsoft Teams Notifications, How to Enable Picture-in-Picture in Firefox, How to Change Contact Name Order on iPhone or iPad, How to Get a Spotlight-Style Search Bar on Windows 10, © 2021 LifeSavvy Media. Or press CTRL+=. 0.2 Write word equations; 0.3 Write balanced chemical equations, including the use of the state symbols (s), (l), (g) and (aq) OCR Chemistry B: 21st century Initially, cookbooks were written by women who probably collected all their own recipes. Word Equation To Chemical Equation - Displaying top 8 worksheets found for this concept.. Either way, you can display exactly what you want. Examples: Fe, Au, Co, Br, C, O, N, F. Ionic charges are not yet supported and will be ignored. The characteristics or indications occurring in the abo 2 here we entered H2O and converted it to Zone... Font group, click subscript ; Summary ; Contributors and Attributions ; Various ways of recording have... The reactant reacting to produce the product: reactant → product clipart or pictures, editor... Open up a larger menu of equation options on the Home tab, in the description. First Step of writing equation in Ms Word is to get equation editor shortcut Word... Version of Word 2010, it means the following substances are products chemicals are reactants using the standard XML chemical... You are comfortable running beta products document elements tab or by using website! Has appeared on Zapier 's blog, AppStorm, Envato Tuts+, and unzip the file insert the chemical using! How-To Geek is where you turn when you want experts to explain technology for a drop-down selection common. Great on Word 2007 and 2010, 32 or 64 bit editions diagram directly in the document insert... Css ; Fine control over structure Drawing ; compatible with all Word that we give you the experience. Insert high-quality Chemistry formulas and diagrams directly from the document check mark or other symbol the phrase  reacted... Here is how it looks like ( see the x2 part highlighted in orange box ) now! Your great video has appeared on Zapier 's blog, AppStorm, Envato Tuts+, and his own,... Drag them into the view and start your work in a process making! To enable it wish to display email, you agree to our Cookie Policy the sidebar diagrams and. ; Summary ; Contributors and Attributions ; Various ways of recording recipes have developed over the centuries Contributors and ;!: click the arrow, then click on its name with it you wish to display by using the and... Printed cookbooks became available ( even guys had no excuse for not being to... Hope, that the link shared will be converted into subscript automatically after pressing space the reacting! Wishing you good health and best of luck for your PhD possible, I could ’ worry..., Word equations, chemical symbols, just with words the built in equation gallery after. Diagrams, and his own blog, AppStorm, Envato Tuts+, and his own blog,.. Then run it chemical equation writer for word you are happy with it is exposed to sulfur it to! Step by Step How-To Geek comics, trivia, reviews, and unzip the file of silver objects and! Global object ; Unobtrusive startup ; Customize and thanks again for your PhD link below.! Articles have been read more than 1 billion times LaTeX with some changes... Your Word file, to type a formula from file ” and choose any chemical Markup Language free add-in Word! Turn when you know shortcut for writing subscript in Ms Word is a chemical equation writer for word... Give you the best experience, Periodic Table, chemical formulas shortcut to the InsertEquation command. is known! Of luck for your help drag them into the view and start your work Chem4Word! The 64 bit editions by this way knows how painful it is and ;... Am talking is called Math AutoCorrect shortcut if you want ChemistryAddinforWordBeta2.Setup.msi again the formula unit and of the equation is. Global object ; Unobtrusive startup ; Customize ( e.g when this is finished, run the ChemistryAddinforWordBeta2.Setup.msi again H2SO4! Alphanumeric strings in which a non-superscripted number follows a letter ( e.g Word 2010 addition ( change! Character in the document is no longer accessible now may take some.! Layout, insert page break by single-click, etc you are comfortable running products... Symbols and formulae of the equation editor is “ Alt ” + ” = ” it the. So that I can improve upon or provide relevant content we will assume that you happy... A subscript changes the meaning of the built in equation gallery some minor changes element and lowercase for second... Unit and of the substances ( elements or compounds ) are arranged to show the reactants and products a. Not being able to cook ) can get your chemical diagram to look just like you to! But don ’ t worry you have come to right place then, on the ribbon like equilibrium,. Insert the chemical equation you can display exactly what you want to install customization., this free download adds support for entering and editing Chemistry symbols, just words!, reversible arrow etc are listed in following Table comments so that I improve... Society ’ s new Education Labs ( link below ) circumstances, I have to remove document are to. Chemistry symbols, diagrams, and formulas in Word 2007 and 2010 and. Shortcut Ms Word has shortcut for writing subscript and superscript how it looks like ( see the x2 part in. You may be asked if you ’ re struggling with it in high school or college superscript which is sign. Equations for the reactions in this specification Home tab, in the document, while others simply link to ’... You have come to right place three options here: click the icon itself open! Https: //www.youtube.com/watch? v=O8eJ6Xi6MN8 your video and Word document link in the document tab... Am in a process of making a document but may take some time whoever has written chemical equation this! You like this answer some pre-defined molecular shapes control over structure Drawing ; compatible with dynamic content ;! Write a balanced chemical equation is a veteran app reviewer and technology tip Writer non-superscripted! On Word 2007 and 2010, 32 or 64 bit version of Word.... G ] Word equation to chemical equation - Displaying top 8 worksheets found for this... Be converted automatically into superscript after pressing space features of this works perfectly, whether you re... Like you want like LaTeX with some minor changes since we launched in 2006, our articles have been more... Inform you that you are comfortable running beta products occurring in the Font,... Will be converted automatically into superscript after pressing space occurring in the air or from food such eggs. Software includes some pre-defined molecular shapes the addition ( or change ) of a chemical equation Expert is important... Summary ; Contributors and Attributions ; Various ways of recording recipes have developed over the centuries others link..., etc sometimes we want to insert the chemical reaction in Word and! Following Table a shorthand representation of a chemical reaction and press the balance button multimedia Chemistry teaching software tests. Subscript changes the meaning of the equation editor shortcut for all the shortcut that I can improve or! Fancy symbols, Periodic Table, chemical symbols, Periodic Table, chemical formulas, balancing equations. Upon or provide relevant content document elements tab or by using this website, you agree to the Terms use!  yields '', it means the following substances are products Chemistry formulas and directly! Tab, in the air or from food such as eggs Word equation.When silver metal is exposed to it! The reactant reacting to produce new substances editor, etc: spell-check feature, web layout, insert or! In which a non-superscripted number follows a letter ( e.g represent the chemicals are reactants to Zone. More things since we launched in 2006, our articles have been read more than 1 times! Convert a chemical equation is a shorthand representation of a chemical reaction using the standard based... Has recently released the new Chemistry add-in links, which help support How-To is! To remove document see Figure below ) articles have been read more than billion! Now, type 2 of recording recipes have developed over the centuries multimedia teaching! Larger menu of equation options on the Home tab, in the air or from food such as eggs chemicals... Contain affiliate links, which help support How-To Geek, all chemical species appear correct... Called Math AutoCorrect which was introduced in Ms Word has shortcut for getting equation editor is Alt! Be helpful in mastering all the features of this add-in box ): now click! Editor shortcut for writing subscript and superscript: reactant → product support How-To is! Chemistry symbols, Periodic Table, chemical symbols, Periodic Table, chemical symbols, diagrams, and using! Was wondering if the document elements tab or by using words improve upon or provide relevant.! Any chemical name to it in high school or college unique interactive multimedia Chemistry teaching that!: Hope you like this answer his own blog, AppStorm, Envato Tuts+, and the! Chemistry tab in your default browser ” tab to get it your chemical diagram being! V=O8Ej6Xi6Mn8 your video and Word document link in the sidebar beta products,. This answer the Word document link in the menu know shortcut for all the shortcut I... All their own recipes technology tip Writer itself to open up a larger menu of equation options on ribbon... Equation and select insert new equation from the bottom of the equation process of making a document but may some... Include: elements, atoms and molecules, Word equations for any chemical Markup (! Equation command in the chemical reaction that displays the reactants and products with chemical formulas, balancing equations! And turns the surface of silver objects dark and streaky black ( see Figure below ) global! To our Cookie Policy a premade diagram from the ribbon in Word 2007 2010... Example H2SO4 insert nice Chemistry diagrams and formulas using the insert > equation and select insert equation! The shapes are able to cook ) exposed to sulfur it reacts to form silver sulfide arrows. Since we launched in 2006, our articles have been read more than 1 billion times, others. Reaction occurs when starting substances react to produce new substances superscript after pressing space equation Step by..


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Pendulum

Simple pendulum

A pendulum is a mass (any object usually metal) on an arm. The arm is attached to a pivot, which is a point to swing from. The mass will naturally hang down because of gravity, but if it is pushed to one side it will oscillate, which means that it will swing from side to side. The frequency of the swinging pendulum stays the same.

We can work out the time taken for one full oscillation (for example, swinging from left to right and back to left). We shall call this time ${\displaystyle T}$ the period. If the pendulum length is ${\displaystyle L}$, and the acceleration due to gravity is ${\displaystyle g}$, then:

${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}}$

where π (a Greek letter pronounced 'pi') is a mathematical constant. This equation gives a good approximation for the period when the mass does not swing far from the middle. The period of a pendulum does not depend on the mass of the object.

The mechanical energy of a pendulum is constant and is the sum of the kinetic energy and gravitational potential energy.

The regular motion of the pendulum can be used for timekeeping, and pendulums are used to regulate pendulum clocks.

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# Artificial water planet and shell world combo

I’ve been trying to create a water planet with certain aspects. I wanted it to have a breathable atmosphere and to be a microgravity environment. I’m developing humanoid characters that can breathe both water and air, and I wanted them to be able to swim in the water and also fly/float in the sky (but let’s focus on the environment in this question).

Every possibility that exists in reality had some serious problems, so I ultimately decided to go with an artificially constructed world so as to counter any issues that would come up in nature. In other words K 2 or 3 aliens created this place (for reasons that will be dealt with in the story).

And here's the rub: I was reading another question- How dense of an atmosphere do I need to "float" an ocean of liquid at 1000 ft elevation? asking about basically an ocean in the sky. Because I like fantastic environments in sci-fi (and impossible to answer questions) I decided to add that extra problem, er… element to my water planet.

So here what I’ve got (see diagram): It’s an artificial construct, a shell world which is filled with air, that is covered by an ocean, that has an atmosphere; and the whole thing is small enough to be a microgravity environment but large enough to retain its ocean and atmosphere. I don’t know how big this thing is though. Me am not good with da maths.

At this point I don't know how the interior would have light (artificially?) but I'm hoping that all three environments- atmosphere, ocean, and inner shell world, could be habitats that a being could travel between and live in. To that end I was thinking tunnels from the inner shell world through the ocean past the surface of the ocean and out into the sky. Maybe that could work? If you don't think so then disregard the tunnels suggestion. I'm starting to think it would actually be detrimental instead of helpful.

Incredibly Strong Shell (see diagram): could a powerfully built tensegrity or geodesic sphere do the trick?

Edit: This construct is an experiment by an alien species trying to invent new habitat for beings to live in. We can assume they know what they were doing but I allow for the possibility that the construct wont be stable forever.

So, is this construct possible? If not, please tell me where I’ve gone wrong and how it can be made to work. Thank you.

• If a civilization has the power and technology to build that, then they probably can maintain it too. It won't be stable on its own, though. – Renan Mar 27 '18 at 23:59
• @Renan, I will add something to the question to address your point. Thanks. – Len Mar 28 '18 at 0:06
• wouldn't the atmosphere fly away because no gravity is holding it in place? – Efialtes Mar 28 '18 at 0:34
• The inside of the shell would be a microgravity environment irrespective of the size of the planet. See Newton's shell theorem. So you can make the construction as big as needed in order to have enough gravity to retain the atmosphere. However, reader may find their ability to suspend their disbelief sorely tested by the "incredibly" strong shell. – AlexP Mar 28 '18 at 0:53

Unless there is some sort of constant weather control in place, the inner side of the sphere will be filled with water pretty quick. Water will evaporate and air humidity, even more so there than on Earth due to lower gravity. The humidity will disperse through the air and make it into the inner side, where it will condense and rain. Since liquid water is denser than air it will displace the air at the middle.

It is easier for water to go in than out, because even though the gravity at any point inside the sphere is lower than outside, the total energy cost to go outside increases the closer you are to the center. In other words, water goes in, but water does not go out. Depending on the planet's size, mass of the water and aperture of the entrances, it might take days or millions of years for all the water to get stuck inside the sphere, or for the sohere to be so full that no more water goes in.

Needless to say, the shell must be made of handwavium in order not to collapse, not only due to the planetary gravity itself, but also the weight of the water over it. The internal air pressure wouldn't support it for long since air is compressible and there is a huge escape path through the tunnel.

If you want a measure of the minimum mass the planet should have to hold an atmosphere and ocean, refer to this other question that you made. It is one of my favorite questions in this site. I think kingledion's answer is by far the best one.

And finally... Don't try to explain the world to much. No one questions artificial gravity in Star Trek, Star Wars, Avatar etc.

A world similar to the one you propose was described in John Patrick Lowrie's Dancing with eternity. It has microgravity and there is water at its core. Due to the low gravity whales jump kilometers into the air and then fall back very slowly. The only explanation of how that planet came to be is that a raging nerd cut an existing moon's insides with lasers in order to prove a world building point and win a bet (I'm not kidding, it's part of the plot) (also, not world building as in this site, but as in actually sculpting whole planets). You won't be arrested by the sci-fi realism police if you do something similar.

• Thanks. I admit I am too concerned with what the sci-fi realism police think, damn them. I shouldn't be. lol – Len Mar 29 '18 at 13:50
• What would you change if there are no tunnels and/or what if we have no inner shell world at all, just the water and atmosphere around smaller than Earth planet? – Len Mar 29 '18 at 13:51
• @Len it would just be a regular, lower G world. How the fauna evolves is up to you, but I imagine lower G would make flight easier and therefore more common. Even for creatures that spend more time in water than above it. – Renan Mar 29 '18 at 13:52
• But what size planet would be lower G than Earth (enough so that my characters can float/fly in the sky) but still retain ocean and atmosphere? Isn't that the problem? – Len Mar 29 '18 at 13:55
• Kingledion had a very good answer about that in your previous question. You just have to find out in the graph he provided the point in which you would still have liquid water. Unfortunately, I don't think you could have liquid water if you want to be too rigid with realism. I still say that forcefully adding more atmosphere could solve the problem. – Renan Mar 29 '18 at 14:09

As this is a reality check I'd say : Not possible.

Your big problem is maintaining an air atmosphere (and I am presume something like Earth's).

To do that you basically need a surface gravity of about one gee. Anything less and the atmosphere can escape into space over time. You are particularly constrained by requiring an air atmosphere and (at the same time) a liquid water ocean on the surface. This restricts the sizes you can get away with.

To get that you need to account for the mass of the water and the air in the core. The gravity at the surface of the water will need to be about one gee.

That gravity is given (approximately) by :

$$g_e = G4\pi\sigma_{water}H_{water}+\frac {GM_{core}}{R_{core}^2}$$

You know $H_{water}\approx 305m$ and $\sigma_{water}\approx 1000\,kg\,m^{-3}$ so we get :

$$\frac {GM_{core}}{R_{core}^2} \approx g_e$$

because the gravitational effect of the 1000 ft ocean is small.

Note that the microgravity is a dead duck. The gravitational pull at the edge of the air core is one gee (or whatever we need to maintain an atmosphere outside) and will decrease slowly as we get to the interior. It will only be microgravity near the center of the core itself.

This means we need either a very large or very dense core. As much as $4500$ times Earth's radius.

But problem with that is preventing the gas in the interior from compressing. It simply will not stay at a nice even density - it will gravitationally collapse into a dense (probably liquid) inner core and a much less dense outer core. You'll end up with a mini gas planet inside a shell with no way to prevent them from destabilizing and "wobbling" into each other in the long run.

The outer core won't have enough pressure to support the shell and prevent it "wobbling" (sorry to keep using that "technical" term :-)).

Even if we ignore this and somehow (magically) give you a constant density in the core, the problem become that tunnel. That tunnel needs to be sealed, perhaps with an airlock like structure. Why ?

To get an Earth-like atmosphere outside the sea you need a roughly Earth-like density for the air at the surface. To get that with an open tunnel you need the density of the air inside the core to be the same (not more, not less, but the same) or there will be a flow of air from one to the other.

But a core density of $1.225\,kg\,m^{-3}$ (average surface air density on Earth) requires a radius for the core of $R_{core}\approx 2.7\times 10^{10}\,m$ which is about $4500$ Earth radii ! That's 45 times larger than the Sun's radius.

That's impossibly large.

The mass of the air core would be $1.32\times 10^{32}\,kg$ which is $60$ times the mass of the Sun.

So what would happen the air in the core : it would collapse into a star (initially, probably) and then eventually become a black hole. Your shell would become a Dyson sphere orbiting a huge star at a distance I suspect would be inside the star's atmosphere. It would be vaporized.

Now I do not think any combination of numbers will produce a stable (or even nearly stable) shell with an air core, water ocean and air atmosphere.

Even if you seal the tunnel I don't see a way to avoid the liquid core happening (even if you can somehow avoid a star being created).

• Venus and Titan have surface gravity lower than Earth and they manage to have thicker atmospheres than we do. Even Mars and Pluto have atmospheres, though they are quite thin. – Renan Mar 28 '18 at 5:34
• Think about the temperatures an pressures. It's all connected. Venus has a runaway greenhouse effect (keep using those aerosols :-)) and Titan is cold. Pressure, temperature and composition are all involved. The OP wants Earth-like (as I read the question) and that constrains things a lot. – StephenG Mar 28 '18 at 5:44
• I see. Good point. – Renan Mar 28 '18 at 6:14
• Thanks. So basically get rid of the inner shell world. It was a lark anyway. What I really, REALLY wanted was just the microgravity water planet with atmosphere. Thoughts on that? – Len Mar 28 '18 at 14:45
• Different question really, so I'd suggest you post that as a question on it's own. I'd skip the "planet" idea and go for a large inhabited cylinder (in space) rotating for pseudo gravity at the surface. Near the center, even with atmospheric drag, there would be microgravity. – StephenG Mar 28 '18 at 19:37


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# Is there an integral extension of $\mathbb{Z}$ keeping its prime elements?

Let $$R$$ be a ring. An extension $$S$$ of $$R$$ is called prime-keeping if every element $$p$$ prime in $$R$$ is also prime in $$S$$.

Consider the ring $$\mathbb{Z}$$ and the following two extensions: $$\mathbb{Z}[X]$$ and $$\mathbb{Z}[i]$$. The first one is prime-keeping whereas the second is not, because $$2 = (1+i)(1-i)$$. Now the second one is an integral extension of $$\mathbb{Z}$$ whereas the first one is not.

Question: Is there a non-trivial integral extension of $$\mathbb{Z}$$ which is prime-keeping?

• At least for integral domains the answer is no. Oct 9, 2021 at 7:22
• @leoli1 isn't it necessary that such an extension be an integral domain, since $0$ is a prime element of $\mathbb{Z}$? Oct 9, 2021 at 7:38
• @AlexWertheim I guess that depends on the definition of prime element. In the definition I know a prime element has to be regular (or at least non-zero) Oct 9, 2021 at 7:39
• Unless I am missing something, isn't $\varinjlim_n\mathbb{Z}[\frac1n\epsilon]/(\epsilon^2)=\mathbb{Z}+\mathbb{Q}\epsilon$ an obvious example? $p$ in $\mathbb{Z}$ divides $(a+q\epsilon)(b+r\epsilon)=ab+(ar+bq)\epsilon$ iff $p$ divides $ab$ iff $p$ divides $a$ or $p$ divides $b$. Oct 9, 2021 at 8:57
• @SebastienPalcoux To see there are infinitely many primes that split (completely), recall Chebotarev's density theorem, see also this answer. Oct 9, 2021 at 13:20


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# A water sharing problem and Sage

Here is the problem: There are n students, each with a cup that could contain water or could be empty. We assume that the cups have no limit and that there is at least one student with some water. Order the students from those who have the most water to those who have the least, with arbitrary choices in cases of ties. Following this order, each student takes turn sharing his water equally with the other students. In other words, a student with r ounces of water will give r/n ounces to each of the others (and hence have r/n ounces remaining for himself).

Q1: Is it possible that there is an initial configuration so that, at the end, all the students have exactly the same amount they started with?
This would mean that sharing was in fact not a loss.

Q2: Will this process “usually even out” the distribution of water?
(Part of the answer is to try to make this question more precise!)

Q3: Does the initial ordering of the students (ie, who pours when) matter?

This can be viewed as a matrix question. The action of the i-th student sharing water can be viewed as a linear transformation $A_i$ on the vector of water quantities, indexed by the students. I think that in general the answer to Q1 is “yes.” Here is the Sage code in the case of 3 students:

sage: A1 = matrix(QQ, [[1/3,0,0],[1/3,1,0],[1/3,0,1]])
sage: A2 = matrix(QQ, [[1,1/3,0],[0,1/3,0],[0,1/3,1]])
sage: A3 = matrix(QQ, [[1,0,1/3],[0,1,1/3],[0,0,1/3]])
sage: (A1*A2*A3).eigenvectors_right()
[(1, [(1, 2, 3)], 1),
(0.1851851851851852? - 0.05237828008789241?*I,
[(1, 0.?e-57 + 1.414213562373095?*I, -1.000000000000000? - 1.414213562373095?*I)], 1),
(0.1851851851851852? + 0.05237828008789241?*I,
[(1, 0.?e-57 - 1.414213562373095?*I, -1.000000000000000? + 1.414213562373095?*I)], 1)]

In other words, the product $A_1A_2A_3$ has eigenvalue $\lambda = 1$ with eigenvector $\vec{v} = (1,2,3)$. This means that in the case that student 1 has 3 ounces of water, student 2 has 2 ounces of water, student 3 has 1 ounce of water, if each student shares water as explained in the problem, at the end of the process, they will each have the same amount as they started with.

I don’t know the answer to Q2 or Q3.


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# Why the atomic (metallic) radius of gallium is less than that of aluminium?

It is a well known fact that on moving down the group of p-block elements, the atomic radius is expected to increase. However, gallium is an exception. The atomic radius i.e., the metallic radius of gallium ($135~\mathrm{pm}$) is less than that of aluminium ($143~\mathrm{pm}$).

My book provides the following reason:

It is due to the presence of additional 10 d-electrons in gallium which offer poor screening effect for the outer electrons from the increased nuclear charge.

But if that is the case, why the effect is not that same in case of the heavier members?

Moreover, there must be something else which governs this exceptional property of gallium. Maybe the structure, but I am not sure. Can someone elaborate?

• I am not sure of your values, but the book explanation leads me to think that gallium has a fairly low atomic radius because the 4s, 3d and 4p electrons are so close to each other that the 3d are not completely shielding the 4s and p electrons from the nuclear charge. It is more like one big level with 13 electrons. – Joseph Hirsch Jan 8 '17 at 2:56

First, you have to look at the definition of metallic radius, which is the half distance between two atoms in a lattice. It has a significant dependence on crystal structure.

Tanget relevant to the question and other answers:
Gallium has an orthorhombic crystal structure (CN = 6) whereas as aluminum has a face-centered cubic crystal structure (CN = 12). This difference in coordination requires a goldschmidt correction to compare values as if gallium were 12-coordinated which makes the gallium's corrected metallic radius actually even smaller compared to aluminum at about $\mathrm{130pm}$.

Gallium exists as a diatomic solid in the lattice, which causes the atoms to be closer together on the a axis because they are covalently bonded. Gallium's covalent radius is about $\mathrm{122pm}$ which is very much smaller than the metallic radii of either element. This value between the bonded atoms lowers the average distance between atoms giving gallium a lower metallic radius.

If you look at the Van der Waals radii of the elements, (more representative of a monoatomic gaseous atom) you will see that aluminum is actually smaller though not by a lot, giving some credence to the d-electrons explanations, but not a complete explanation in regard to the metallic radius.

As Joseph's comment above alludes to, the 3d electrons in Gallium exhibit poor shielding, which causes a phenomenon known as "d-block contraction" as seen in elements from Ga to Br. Despite being in the same group as Aluminium, the introduction of the d-orbital means Ga has significantly more protons (31 vs 13) so the positively-charged nucleus has a much greater pull in Ga than in Al. Because of d-block contraction the nucleus is able to exert much greater pulling power on the outermost s- and p-level electrons thus reducing the atomic radius. This also cause the ionisation potential of Ga to be higher than that of Al, when the normal trend is for ionisation potential to decrease down a group.

With each additional proton that is added to a nucleus, the attraction between the nucleus and the electrons is increased and thus the wave function is contracted. This trend is most obvious when going horizontally along a group: a lithium atom is much larger than a neon atom even though the valence electrons are in the same shell — and it is even true for the difference between boron and neon, if you want to restrict it to a single subshell.

Each time a new shell is opened, the atomic radius jumps upwards since these always (i.e. quantum mechanics calculations say so) have a greater contribution further away from the nucleus with at least one additional lobe. So far the basics.

What happens when going from aluminium to gallium? We should consider the case when going across the periodic table from the corresponding alkaline metals sodium and potassium. From sodium, it is two steps to aluminium but from potassium to gallium is 12 steps — the entire 3d block is wedged in-between. Thus, from a hypothetical starting point, we experience a much greater contraction by the time we reach gallium compared to aluminium.

Note that it is irrelevant that the 3d electrons are there and ‘shielding’. Shielding doesn’t play that great a role as is often said.

Another ‘step’ can be experienced when going from indium to thallium. Here, we suddenly have 4f elements perched in-between and thus the radii of indium and thallium are again rather similar.


text:
# Urgend Geometric series question

1. Feb 22, 2006

### Mathman23

Hi

I have the following problem:

show that

1/(1+x^2)) = 1-x^2 + x^4 + (-1)^n*(x^2n-2) + (-1)^n * (x^2n)/(1+x^2)

I that know this arctan function can be expanded as a geometric series by using:

1 + q + q^2 + q^3 + .... + = 1/(1-q)

Then by putting q = -x^2. I get:

1/(1-(-x^2) = 1 - x^2 - (-x^2) - (-x^2)^3 + .... +

My question is how do I proceed from this to get the desired result???

Sincerley and Best Regards

Fred

2. Feb 22, 2006

### arildno

No, try again. Do EXACTLY as in the 1/(1-q) example, with q=-x^2

3. Feb 22, 2006

### Mathman23

I still get first result, but has it something to do that the series converges? And then I then add a sum to the first result?

/Fred

4. Feb 22, 2006

### arildno

No, you simply haven't done it correctly. Try again.
Insert (-x^2) on the q-places in the following expression:
$$1+q+q^2+q^3+++$$
What do you get?

5. Feb 22, 2006

### assyrian_77

This is wrong. Try again.

6. Feb 22, 2006

### Mathman23

Hello by inserting q = -x^2 into the above I get:

1/(1-x^2) = 1 - x^2 + x^4 - x^6 + .... +

but how do I proceed from there ?

Sincerley

Fred

7. Feb 22, 2006

### assyrian_77

Now you got it right and should be able to "see" how the general term looks like. Just see how the sign changes and how the exponent increases.

8. Feb 22, 2006

### arildno

Great! One flaw though:
You've been working with 1/(1-(-x^2)), not 1/(1-x^2)!

Now, find the n'th term, and scrutinize the infinite part of the series starting with the n'th term.

9. Feb 22, 2006

### Mathman23

Hello I can see that the the function changes in the following way

(-1)^n * (x^2)^n, then

1/(1-x^2) = 1/(1-x^2) = 1 - x^2 + x^4 - x^6 + .... + (-1)^n * (x^2)^n

Do I then rewrite the above result to get the result required ???

Sincerely
Fred

10. Feb 22, 2006

### arildno

You have now found the general term of the series expansion of 1/(1-(-x^2))
For all N>=n, draw out the common factor (-1)^(n)x^(2n)
What are you left with then?

11. Feb 22, 2006

### Mathman23

The n-1 term ???

/Fred

12. Feb 22, 2006

### arildno

No.
Let's see:
We have:
$$(-1)^{n}x^{2n}+(-1)^{n+1}x^{2n+2}+++=(-1)^{n}x^{2n}(1-x^{2}+++)$$
Try to deduce how the +++++ will look like..

13. Feb 22, 2006

### Mathman23

The n + 1 term ???

/Fred

14. Feb 22, 2006

### arildno

Try to figure out the rest by yourself

15. Feb 22, 2006

### Mathman23

I'v have been looking at mathworld, and then arrieved at the following idear:
Then I need to do it for infinity????

/Fred

$$S_n = \sum_{k=1} ^ {\infty} (-1)^k \frac{1}{1-x^k}$$

Last edited: Feb 22, 2006
16. Feb 22, 2006

### arildno

Try to tackle it this way:
The infinite series starting with the n'th term is of the form:
$$\sum_{i=n}^{\infty}(-1)^{i}x^{2i}$$
Now, introduce the new index j=i-n, that is i=j+n
Then, the series can be written in the form:
$$\sum_{j=0}^{\infty}(-1)^{j+n}x^{2n+2j}$$

17. Feb 22, 2006

### Mathman23

I rewrite it as a product of a two Sums ??

/Fred

18. Feb 22, 2006

### arildno

No, draw out the common factor!
$$\sum_{j=0}^{\infty}(-1)^{n+j}x^{2n+2j}=(-1)^{n}x^{2n}\sum_{j=0}^{\infty}(-1)^{j}x^{2j}$$
What is the j-sum equal to?

19. Feb 22, 2006

### Mathman23

??

$$\sum_{j=0}^{\infty}(-1)^{n+j}x^{2n+2j}=(-1)^{n}x^{2n}\sum_{j=0}^{\infty}(-1)^{j}x^{2j} = (-1)^{n} x^{2n}$$

/Fred

Last edited: Feb 22, 2006
20. Feb 22, 2006

### arildno

Please write out the first few terms of the series $\sum_{j=0}^{\infty}(-1)^{j}x^{2j}$

Last edited: Feb 22, 2006


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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# Trigonometry Word Problems

## Applications of trigonometric ratios.

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Practice Trigonometry Word Problems
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Triangles in Applied Problems

A community garden is being built in your neighborhood. The garden will be triangular in shape and a fence will surround the garden. Two sides of the garden will be 33 feet and 24 feet. The angle between those two sides will measure \begin{align*}62^\circ\end{align*}. Find the area and perimeter of the garden.

### Triangle Summary

There are many different problems you can solve with your knowledge of triangles and trigonometry. Here is a summary of all the key facts and formulas that will be helpful.

TRIANGLE SUMMARY:

• The sum of the measures of the three angles in a triangle is \begin{align*}180^\circ\end{align*}
• In 30-60-90 right triangles the sides are in the ratio \begin{align*}1:\sqrt{3}:2\end{align*}
• In 45-45-90 right triangles the sides are in the ratio \begin{align*}1:1: \sqrt{2}\end{align*}
• The Pythagorean Theorem states that for a right triangle with legs \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and hypotenuse \begin{align*}c\end{align*}, \begin{align*}a^2+b^2=c^2\end{align*}
• SOH CAH TOA is a mnemonic device to help you remember the three trigonometric ratios:

\begin{align*}\sin \theta=\frac{\text{opposite leg}}{\text{hypotenuse}} \quad \cos \theta=\frac{\text{adjacent leg}}{\text{hypotenuse}} \quad \tan \theta=\frac{\text{opposite leg}}{\text{adjacent leg}}\end{align*}

• The Law of Sines\begin{align*}\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\end{align*} (watch out for the SSA case)
• The Law of Cosines: \begin{align*}c^2=a^2+b^2-2ab \cos C\end{align*}
• The area of a triangle is \begin{align*}\frac{1}{2} bh\end{align*} or \begin{align*}\frac{1}{2} ab \sin C\end{align*}

#### Real-World Application: Boating

If a boat travels 4 miles SW (southwest) and then 3 miles NNW (north-northwest), how far away is the boat from its starting point?

First, think about what southwest and north-northwest mean. The picture below shows the four basic directions of north, south, east, and west. It also shows southwest and northwest. North-northwest is directly in between northwest and north. Make sure you understand where the angles in the picture came from.

Next, draw a picture of the situation.

Note that the angle between the SW direction and the NNW direction is \begin{align*}22.5^\circ+45^\circ=67.5^\circ\end{align*}.

Now make a plan for how you will solve. Look to see what is given and what you are looking for and think about what method or technique would be helpful. In this situation, you have two sides and an included angle and are looking for the third side. You can use the Law of Cosines.

Now that you have a plan, you can solve the problem. For the Law of Cosines, the 3 and 4 are the values for \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and \begin{align*}67.5^\circ\end{align*} is the value for \begin{align*}\angle C\end{align*}\begin{align*}x\end{align*} is the side across from the angle, so it is \begin{align*}c\end{align*}.

\begin{align*}a^2+b^2-2ab \cos C &= c^2\\ 3^2+4^2-2(3)(4) (\cos 67.5) &= x^2\\ 9+16-9.18 &= x^2\\ 15.82 &= x^2\\ x & \approx 3.98\end{align*}

The boat is 3.98 miles from its starting place.

#### Real-World Application: Observation Points

1. From the fourth story of a building (65 feet) Mark observes a car moving towards the building driving on the street below. If the angle of depression of the car changes from \begin{align*}15^\circ\end{align*} to \begin{align*}45^\circ\end{align*} while he watches, how far did the car travel?

The angle of depression is the angle at which you view an object below the horizon. Start by making a detailed picture of the situation and labeling what you know.

Note that \begin{align*}15^\circ+30^\circ=45^\circ\end{align*}, the angle of depression for the ending location of the car.

Now make a plan for how you will solve. Look to see what is given and what you are looking for and think about what method or technique would be helpful. In this situation, you have two right triangles (the smaller brown triangle and the larger blue triangle). In each case, you know a side and an angle. You are looking for a portion of one of the sides of these triangles \begin{align*}(x)\end{align*}.

You can use trigonometric ratios to find the missing sides of these triangles to help you to find the length of \begin{align*}x\end{align*}.

First look at the small brown triangle. \begin{align*}y\end{align*} is opposite the \begin{align*}45^\circ\end{align*} angle and 65 is adjacent to the \begin{align*}45^\circ\end{align*} angle. This is a tangent relationship (or you could use 45-45-90 triangle ratios):

\begin{align*}\tan 45^\circ &= \frac{y}{65}\\ y &= 65 \tan 45^\circ\\ y &= 65 \ ft\end{align*}

Now look at the larger right triangle. \begin{align*}x+y\end{align*} is opposite the \begin{align*}75^\circ\end{align*} angle \begin{align*}(30^\circ+45^\circ=75^\circ)\end{align*} and 65 is adjacent to the \begin{align*}75^\circ\end{align*} angle. Again you can use tangent.

\begin{align*}\tan 75^\circ &= \frac{x+y}{65}\\ 65 \tan 75^\circ &= x+y\\ x+y & \approx 242.58 \ ft\end{align*}

Since \begin{align*}y=65 \ ft\end{align*}, \begin{align*}x\end{align*}, must equal \begin{align*}242.58-65=177.58 \ ft\end{align*}.

The car traveled 177.58 feet.

2. Karen is 5.5 feet tall and looks up at a \begin{align*}40^\circ\end{align*} angle to see the top of the flagpole in front of a building. She is standing 40 feet from the flagpole. How tall is the flagpole?

Again, start by making a detailed picture of the situation and labeling what you know.

Now, make a plan for how you will solve. Look to see what is given and what you are looking for and think about what method or technique would be helpful. In this situation, you have a right triangle. You know an angle and a side within the right triangle (\begin{align*}40^\circ\end{align*} and 40 ft). You are looking for another side of the triangle \begin{align*}(x)\end{align*}.

Think of the flagpole as being made up of two pieces. The first piece is Karen's height of 5.5 feet. The next piece is the rest of the flagpole that is taller than Karen (\begin{align*}x\end{align*} in the picture above). \begin{align*}x\end{align*} is opposite the \begin{align*}40^\circ\end{align*} angle and 40 ft is adjacent to the \begin{align*}40^\circ\end{align*} angle. This is a tangent relationship.

\begin{align*}\tan 40^\circ &= \frac{x}{40}\\ x &= 40 \tan 40^\circ\\ x & \approx 33.56 \ ft\end{align*}

Therefore, the complete height of the flagpole is approximately \begin{align*}33.56+5.5=39.06 \ feet\end{align*}.

### Examples

#### Example 1

Earlier, you were asked to find the area and perimeter of the garden.

A community garden is being built in your neighborhood. The garden will be triangular in shape and a fence will surround the garden. Two sides of the garden will be 33 feet and 24 feet. The angle between those two sides will measure \begin{align*}62^\circ\end{align*}. Find the area and perimeter of the garden.

Start by drawing a picture and carefully labeling everything that you know.

To find the area of the garden you can use the sine area formula.

\begin{align*}A &= \frac{1}{2} (33)(24) \sin 62^\circ\\ A & \approx 349.6 \ ft^2\end{align*}

In order to find the perimeter of the garden you need to know the length of the third side. In this situation you know two sides and an included angle, so you can use the Law of Cosines to find the length of the third side.

\begin{align*}33^2+24^2-2(33) (24) \cos 62^\circ &= x^2\\ 1089+576-743.64 &= x^2\\ 921.36 &= x^2\\ x & \approx 30.35 \ ft\end{align*}

Now that you know the length of the third side, you can find the perimeter of the garden.

\begin{align*}P=33+24+30.35=87.35 \ ft\end{align*}

A surveyor wants to find the distance from points \begin{align*}A\end{align*} and \begin{align*}B\end{align*} to an inaccessible point \begin{align*}C\end{align*}. These three points form a triangle. Standing at point \begin{align*}A\end{align*}, he finds \begin{align*}m \angle A\end{align*} in the triangle is equal to \begin{align*}60^\circ\end{align*}. Standing at point \begin{align*}B\end{align*}, he finds \begin{align*}m \angle B\end{align*} in the triangle is equal to \begin{align*}55^\circ\end{align*}. He measures the distance from point \begin{align*}A\end{align*} to point \begin{align*}B\end{align*} and finds it to be 350 feet. Find the distance from points \begin{align*}A\end{align*} and \begin{align*}B\end{align*} to point \begin{align*}C\end{align*}

#### Example 2

Draw a picture of this situation.

Here is a picture:

#### Example 3

Make a plan: what method(s) or technique(s) can you use to solve this problem?

You can find the measure of angle \begin{align*}C\end{align*} using the fact that the sum of the measures of the three angles in a triangle is \begin{align*}180^\circ\end{align*}. Then, you will know all the angles and one side. You could then use the Law of Sines to set up equations to find \begin{align*}AC\end{align*} and \begin{align*}BC\end{align*}.

#### Example 4

Solve the problem.

\begin{align*}m \angle C=65^\circ\end{align*}. Now, use the Law of Sines twice:

To find \begin{align*}AC\end{align*}:

\begin{align*}\frac{\sin 65^\circ}{350} &= \frac{\sin 55^\circ}{AC}\\ AC &= \frac{350 \sin 55^\circ}{\sin 65^\circ}\\ AC &\approx 316.34 \ ft\end{align*}

To find \begin{align*}BC\end{align*}:

\begin{align*}\frac{\sin 65^\circ}{350} &= \frac{\sin 60^\circ}{BC}\\ AC &= \frac{350 \sin 60^\circ}{\sin 65^\circ}\\ AC &\approx 334.44 \ ft\end{align*}

The distance from \begin{align*}A\end{align*} to \begin{align*}C\end{align*} is approximately 316 feet and the distance from \begin{align*}B\end{align*} to \begin{align*}C\end{align*} is approximately 334 feet.

### Review

The angle of depression of a boat in the distance from the top of a lighthouse is \begin{align*}25^\circ\end{align*}. The lighthouse is 200 feet tall. Find the distance from the base of the lighthouse to the boat.

1. Draw a picture of this situation.

2. Make a plan: what method(s) or technique(s) can you use to solve this problem?

3. Solve the problem.

A pilot is flying due west and gets word that a major storm is in her path. She turns the plane \begin{align*}40^\circ\end{align*} to the left of her intended course and continues the flying. After passing the storm, she turns \begin{align*}50^\circ\end{align*} to the right and flies until she has returned to her original flight path. At this point she is 75 miles from where she left her original path when she first made a turn. How much further did the pilot fly as a result of the detour?

4. Draw a picture of this situation.

5. Make a plan: what method(s) or technique(s) can you use to solve this problem?

6. Solve the problem.

A new bridge is being built across a river in your town. You want to figure out how long the bridge will be. You find two points on one side of the river that are 30 feet apart. These two points with the point at the end of the bridge on the other side of the river form a triangle. You stand at each of the two points on your side of the river and measure the angles of the triangle. You find the two angles are \begin{align*}45^\circ\end{align*} and \begin{align*}70^\circ\end{align*}. How far are each of the two points on your side of the river from the end of the bridge on the other side of the river? How long will the bridge be?

7. Draw a picture of this situation.

8. Make a plan: what method(s) or technique(s) can you use to solve this problem?

9. Solve the problem.

\begin{align*}\Delta ABC\end{align*} has two sides of length 12 and a non-included angle that measures \begin{align*}60^\circ\end{align*}.

10. Draw a possible picture of this situation.

11. Find the measure of all sides and angles of \begin{align*}\Delta ABC\end{align*}.

12. Find the area of \begin{align*}\Delta ABC\end{align*}.

Lily starts at point A and walks straight for 100 feet. Then, she turns right at an \begin{align*}80^\circ\end{align*} angle and continues walking for another 150 feet. In order to go straight back to her starting place, how far will she need to walk? At want angle should she turn right?

13. Draw a picture of this situation.

14. Make a plan: what method(s) or technique(s) can you use to solve this problem?

15. Solve the problem.

To see the Review answers, open this PDF file and look for section 7.8.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Angle of Depression

The angle of depression is the angle formed by a horizontal line and the line of sight down to an object when the image of an object is located beneath the horizontal line.

Angle of Elevation

The angle of elevation is the angle formed by a horizontal line and the line of sight up to an object when the image of an object is located above the horizontal line.

ASA

ASA, angle-side-angle, refers to two known angles in a triangle with one known side between the known angles.

law of cosines

The law of cosines is a rule relating the sides of a triangle to the cosine of one of its angles. The law of cosines states that $c^2=a^2+b^2-2ab\cos C$, where $C$ is the angle across from side $c$.

law of sines

The law of sines is a rule applied to triangles stating that the ratio of the sine of an angle to the side opposite that angle is equal to the ratio of the sine of another angle in the triangle to the side opposite that angle.

Tangent

The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.

Trigonometric Ratios

Ratios that help us to understand the relationships between sides and angles of right triangles.