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By: Sam Shearman

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# Texting Over the Speaker-Air-Microphone (SAM) Channel

Module by: Ed Doering. E-mail the authorEdited By: Sam Shearman, Erik Luther, Brett Hern

Summary: Texting and instant messaging are familiar and popular communication techniques. Text messages composed of ASCII characters, with each character composed of eight bits, ultimately form a stream of bits transmitted from the source to the receiver. Normally the text message moves through a complex data network, and the user only sees the endpoints of the system. In this project, however, you will be able to follow the text message on its complete journey through a binary ASK transmitter, a speaker-air-microphone (SAM) channel, and a binary ASK receiver.

 This module refers to LabVIEW, a software development environment that features a graphical programming language. Please see the LabVIEW QuickStart Guide module for tutorials and documentation that will help you: • Apply LabVIEW to Audio Signal Processing • Get started with LabVIEW • Obtain a fully-functional evaluation edition of LabVIEW

## Note:

Visit LabVIEW Setup to learn how to adjust your own LabVIEW environment to match the settings used by the LabVIEW screencast video(s) in this module. Click the "Fullscreen" button at the lower right corner of the video player if the video does not fit properly within your browser window.

## Summary

Texting and instant messaging are familiar and popular communication techniques. Text messages composed of ASCII characters, with each character composed of eight bits, ultimately form a stream of bits transmitted from the source to the receiver. Normally the text message moves through a complex data network, and the user only sees the endpoints of the system. In this project, however, you will be able to follow the text message on its complete journey through a binary ASK transmitter, a speaker-air-microphone (SAM) channel, and a binary ASK receiver.

This project depends on successful completion of the two prerequisite projects, Speaker-Air-Microphone (SAM) Channel Characterization and Binary ASK Transmitter.

## Objectives

1. Implement an ASK demodulator based on an envelope detector, a noncoherent detector
2. Appreciate the importance of a message preamble for successful receiver operation
3. Translate the ASK receiver block diagram into a LabVIEW block diagram

## Deliverables

1. Summary write-up of your results
2. Hardcopy of all LabVIEW code that you develop (block diagrams and front panels)
3. Any plots or diagrams requested

### Note:

You can easily export LabVIEW front-panel waveform plots directly to your report. Right-click on the waveform indicator and choose "Export Simplified Image."

## Setup

1. LabVIEW 8.5 or later version
2. Modulation Toolkit 4.0 or later version
3. Computer soundcard that supports full-duplex operation
4. Speaker
5. Microphone
6. Two computers (optional)

Refer to the following textbooks for additional background on the project activities of this module; see the "References" section below for publication details:

• Carlson, Crilly, and Rutledge -- Ch 14
• Couch -- Ch 3 and 5
• Haykin and Moher -- Ch 7
• Lathi -- Ch 13
• Proakis and Salehi (FCS) -- Ch 10
• Stern and Mahmoud -- Ch 5

## Prerequisite Modules

Complete the lab projects Speaker-Air-Microphone (SAM) Channel Characterization and Binary ASK Transmitter before you begin this project.

If you are relatively new to LabVIEW, consider taking the course LabVIEW Techniques for Audio Signal Processing which provides the foundation you need to complete this project activity, including: block diagram editing techniques, essential programming structures, subVIs, arrays, and audio.

## Introduction

Figure 1 shows the hardware setup for this project. One computer running a "text sender" LabVIEW VI translates a text message into a bitstream, modulates the bitstream as an ASK signal, and plays the signal to a speaker. A second computer running a "text receiver" VI listens to the microphone signal, records and demodulates the ASK signal, regenerates the transmitted bitstream, and translates the bitstream back to text. The text sender and receiver VIs can run in parallel on a single computer to simplify development.

The frequency response and bandwidth of the speaker-air-microphone (SAM) channel were measured in an earlier project, Speaker-Air-Microphone (SAM) Channel Characterization. As discussed in the Binary ASK Transmitter project, the bit rate and pulse shape can be adjusted to place the transmitted signal in various sub-bands or to fully occupy the available channel bandwidth. The channel bandwidth can be allocated in a number of different ways for this project. For example, use the full channel bandwidth to minimize the amount of time to transmit a message. However, transmitting messages simultaneously between two computers requires two different transmitter carrier frequencies and careful choice of bit rates and pulse shapes to ensure the signals remain contained in a sub-band of the channel.

The text message to be transmitted is a collection of ASCII characters. ASCII, the American Standard Code for Information Interchange, defines a mapping between familiar alpha-numeric characters (as well as other types of characters) and 7-bit binary patterns. For example, the five character sequence "hello" maps to the bit sequence 1101000 1100101 1101100 1101100 1101111. See the Wikipedia article on ASCII for ASCII mapping tables and further information.

LabVIEW offers a complete set of tools to manipulate and process text, including methods to translate text to bit sequences and vice versa. Swapping the random number generator currently serving as the message source in the transmitter from the Binary ASK Transmitter project with a text-to-bitstream converter quickly creates the necessary text sender for this project.

Figure 2 shows a high-level block diagram of the ASK receiver to be constructed in this project.

The recorded microphone signal serves as the receiver input. The receiver filter is a bandpass filter centered at the transmitter's carrier frequency with a bandwidth selected to match the transmission bandwidth. The receiver filter therefore admits the signal of interest but removes out-of-band noise and signals in adjacent bands.

The ASK signal spectrum contains a strong carrier component, making noncoherent detection feasible. Coherent detection requires carrier synchronization, a more complex approach. Noncoherent detection, on the other hand, can be simply implemented using an envelope detector. The envelope detector recovers the baseband signal. The Figure 3 screencast video discusses the envelope detector principles of operation, implementation by a lowpass filter followed by a rectifier, and choice of lowpass filter corner frequency based on bit rate.

The normalizer, the third block in the sequence, shifts the baseband signal to eliminate the DC bias and scales the signal to the range ±1. The normalizer can be considered a type of automatic gain control (AGC) to compensate for variations in speaker signal strength and microphone gain settings.

Up to this point the baseband signal can be considered an "analog" signal, even though it is in reality a discrete-time sampled signal. The sampler selects a single sample point from the baseband signal once each bit interval. The sampling process itself is very simple, but how does the sampler know when a bit interval has elapsed? Fortunately a hidden yet vital component in the sampler block called the bit synchronizer extracts timing pulses directly from the baseband signal, provided the signal meets certain requirements.

The Figure 4 screencast video discusses the bit synchronizer principles of operation and implementation considerations.

As discussed in the previous video, the bit synchronizer's bandpass filter can be considered a high-Q resonator tuned to the bit rate, where Q stands for quality factor, the ratio of the bandpass filter's center frequency to its bandwidth; high Q indicates narrow bandwidth. The resonator produces sustained oscillations when the input bitstream contains approximately equiprobable ones and zeros. The high Q value assures that the resonator will continue to oscillate for a short time even when the bitstream has occasional runs of constant values. A high-Q resonator possesses inertia -- once oscillations have started, the resonator will continue to oscillate for a time, even without excitation. However, just as in any system with inertia, an inactive resonator requires excitation for a significant period of time before oscillation reaches a useful amplitude.

Now consider a receiver waiting for a signal, and suppose the transmitter only generates nonzero signals when it is actually sending a message. If the bitstream can only be sampled once the bit synchronizer "revs up," so to speak, how can the first few bits of the message be correctly sampled? In fact, the first few bits of the message could not be recovered this way. However, the transmitter is free to transmit additional bits before the message bits -- a preliminary bit sequence called a preamble.

The Figure 5 screencast video discusses the design of the preamble to accomplish two goals: (1) to "wake up" the bit synchronizer (more formally called channel seizure), and (2) to facilitate frame synchronization, the process of recovering meaningful symbols (ASCII characters, in this case) from the stream of ones and zeros.

Consider once again the receiver block diagram of Figure 2. The threshold comparator (also called the decision device) follows the sampler and compares each sampled signal value to zero. A sampled signal value higher than the threshold is declared a binary 1, and a value lower than the threshold is declared a binary 0. The threshold comparator completes the bitstream regeneration process; in the absence of noise, the bitstream produced by the comparator is identical to the transmitted bitstream.

The last block in the receiver retrieves ASCII characters from the bitstream. The earlier Figure 5 screencast video briefly discussed the need for frame synchronization, which is now considered in more detail. Suppose the message preamble consisted only of the alternating one-zero pattern necessary to start the bit synchronizer. Certainly the total length of the preamble is known, so in principle it would seem that the first message bit could be determined after some well-defined number of cycles after the bit synchronizer starts up. The problem is that this "well-defined" number of cycles can vary slightly due to noise levels, signal amplitude, and many other variables; the first actual message bit cannot be determined reliably this way.

Also, suppose the individual ASCII bit patterns were all transmitted as one contiguous bitstream by packing all the 7-bit patterns one after another. If the first message could not be determined correctly, then all remaining bits would likewise be interpreted incorrectly. Good system design should be fault tolerant, meaning that an interpretation error on one character should not affect the entire message.

These problems are addressed by inserting framing bits around each ASCII character. The Figure 6 screencast video describes the concepts of framing bits and frame synchronization, and why a sequence of binary ones (also called the steady mark sequence) is an important second part of the preamble.

## Procedure

### SubVI construction

Build the subVIs listed below. You may already have some of these available from previous projects.

Demonstrate that each of these subVIs works properly before continuing to the next part.

### Text sender

Copy the application VI you completed at the end of the Binary ASK Transmitter project to a new VI named TextSender.vi. Be sure to use the version that includes raised cosine pulse shaping as an option.

Modify the message generation section to use util_BitstreamFromText.vi to convert a front-panel text control value into a bitstream.

Design a preamble sequence that satisfies the requirements for bit sync and frame sync described in the introduction. Concatenate the preamble to the beginning of the message bitstream.

Use util_BitstreamToText.vi to display the complete transmitted message in hexadecimal format.

Listen to your transmitted ASK signal and view the power spectrum for several choices of bit rate, carrier frequency, and pulse shape.

The Figure 7 screencast video describes two different ways to form the preamble sequence, and describes how to choose the hexadecimal display modes for text controls and indicators.

### Audio test signals

Record several receiver input signals as audio .wav files as test signals to save development time in the next section. Use sam_GrabAudioDynamic.vi to capture the audio and save to a .wav file using "Graphics and Sound | Sound | Files | Sound File Write Simple.vi" as shown in Figure 8.

Review the ASK receiver block diagram of Figure 2. Assemble your subVIs to create TextReceiver.vi according to the block diagram. Note that several of the blocks on the diagram are composed of two subVIs: (1) The "normalizer" includes regen_ExtractPreamble.vi and regen_NormalizeToPreamble.vi, (2) the "sampler" includes regen_BitSync.vi and regen_Sampler.vi, and (3) the "bitstream to text" block includes regen_FrameSync.vi and util_BitstreamToText.vi.

Remember to test as you build. Start with the input signal, add a processing block, and then add relevant waveform indicators to confirm correct operation before you proceed to the next processing block.

Include the following front panel control values:

• recorded .wav file -- path
• use recorded signal -- Boolean
• fs, sampling frequency [Hz] -- DBL
• fc, carrier frequency [Hz] -- DBL
• Tb, bit interval [s] -- DBL
• adjust sampling instant [% of bit interval] (0%) -- DBL
• receiver filter bandwidth [Hz] -- DBL

The following front-panel indicators, especially the waveform graph indicators stacked vertically in the order listed below, visualize the complete "inner workings" of the ASK demodulation, bit sync, and frame sync operations; these visualizations will greatly improve your understanding of the behavior of each major component of the receiver as well as their interrelationships:

• received signal -- the receiver input signal from the microphone or test file
• received spectrum -- with amplitude calibrated in dB
• receiver filter spectrum -- received filter output signal, with amplitude calibrated in dB and Y-scale to match the received spectrum; output of the receiver filter
• receiver filter -- receiver filter output signal
• baseband -- envelope filter output signal
• preamble -- regen_ExtractPreamble.vi output signal
• normalized baseband -- regen_NormalizeToPreamble.vi output signal
• baseband absolute value -- intermediate signal from regen_BitSync.vi
• bandpass filter -- intermediate signal from regen_BitSync.vi
• thresholded bandpass filter -- intermediate signal from regen_BitSync.vi
• baseband + sampling instants -- envelope detector and "actual sampling instants" from regen_Sampler.vi signals overlaid on the same waveform graph

Also include the following indicators on the front panel, clustered together below the vertical stack of waveform graphs:

• eye diagram -- waveform graph; output of the LabVIEW Modulation Toolkit "MT Format Eye Diagram.vi" to show the eye diagram of the baseband signal produced by the envelope detector
• preamble detected? -- Boolean from regen_ExtractPreamble.vi
• message detected? -- Boolean from regen_FrameSync.vi
• bitstream length -- length of bitstream produced by regen_FrameSync.vi
• bitstream -- Boolean 1-D array produced by threshold comparator
• bitstream (hex) -- text with hex display mode to show output of threshold comparator
• received message -- text produced by util_BitstreamToText.vi
• framing error? -- Boolean 1-D array produced by util_BitstreamToText.vi
• received frames -- Boolean 2-D array produced by util_BitsToWords.vi
• bit rate [Hz] -- reciprocal of bit interval, Tb

The Figure 9 screencast video shows a suggested front-panel layout for the text receiver.

### Testing and demonstration

Debug and test your receiver with your pre-recorded audio .wav files. Once you are satisfied that your receiver works properly, run your text sender as a separate application. Your soundcard is likely full duplex, and so both the text sender and text receiver should be able to access the soundcard simultaneously. You may also like to work with a friend with his or her own computer to make the texting activity more interesting.

You may also want to experiment with full-duplex texting between two different machines. Choose two different carrier frequencies, one for your friend and another for yourself. Be sure to choose a bit rate and pulse shaping so that each of you only uses up to half of the available SAM channel bandwidth. Try relatively long messages and confirm that you can simultaneously send and receive messages. This is the essence of frequency division multiplexing.

## References

1. Carlson, A. Bruce, Paul B. Crilly, and Janet C. Rutledge, "Communication Systems," 4th ed., McGraw-Hill, 2002. ISBN-13: 978-0-07-011127-1
2. Couch, Leon W. II, "Digital and Analog Communication Systems," 7th ed., Pearson Prentice Hall, 2007. ISBN-10: 0-13-142492-0
3. Haykin, Simon, and Michael Moher, "Introduction to Analog and Digital Communication Systems," 2nd ed., Wiley, 2007. ISBN-13: 978-0-471-43222-7
4. Lathi, Bhagwandas P., "Modern Digital and Analog Communication Systems," 3rd ed., Oxford University Press, 1998. ISBN-10: 0-19-511009-9
5. Proakis, John G., and Masoud Salehi, "Fundamentals of Communication Systems," Pearson Prentice Hall, 2005. ISBN-10: 0-13-147135-X
6. Stern, Harold P.E., and Samy A. Mahmoud, "Communication Systems," Pearson Prentice Hall, 2004. ISBN-10: 0-13-040268-0

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58

Introduction This post is long overdue as I have been repeatedly asked to explain code of mine containing these things. As I see increased use of this construct by others perhaps it is past due also. SparseArray objects can behave as functions accepting certain arguments to return internal data or efficiently return data in certain forms. These are known ...

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The following simulates Mathematica's behaviour after using it for more than 24 hrs. MathLinkCallFrontEnd[FrontEndUndocumentedCrashFrontEndPacket[]] Works as advertised! :D

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Sequential With From Daniel Lichtblau's comment there is a new undocumented syntax for With introduced sometime after version 10.1 that allows: With[{a = 0}, {a = a + 1}, {a = a + 1}, a] 2 Delayed With, Block, and Module These scoping constructs support the use of := in definitions which allows one to handle unevaluated expressions. With[{x := 2 + 2}, ...

40

Funny you should ask :), it turns out there is an undocumented use of Integrate that one can leverage to integrate over regions. Unfortunately this does not seem to work for NIntegrate. This usage is better leveraged in conjunction with some other undocumented functions (see here and here). I will show a few examples of how to use this feature. First, let's ...

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InternalPartitionRagged This one has a usage statement! InternalPartitionRagged[Range[14], {3, 5, 2, 4}] {{1, 2, 3}, {4, 5, 6, 7, 8}, {9, 10}, {11, 12, 13, 14}} Note that Length[list] must equal n1 + ... + nk. (* changed the last 4 to 3 *) InternalPartitionRagged[Range[14], {3, 5, 2, 3}] InternalPartitionRagged[Range[14], {3, 5, 2, 3}] InternalS1,...

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InternalLocalizedBlock behaves the same as Block, but it can localize non-Symbols (e.g. f[1], Subscript[x, 0], etc.). For example, InternalLocalizedBlock[{Subscript[x, 0]}, Subscript[x, 0] = 1] (* 1 *) Compare this to Block[{Subscript[x, 0]}, Subscript[x, 0] = 1] (* During evaluation of In[79]:= Block::lvsym: Local variable specification {Subscript[x, ...

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There is the undocumented option Beveled in the FractionBox FractionBox[a, b, Beveled -> True] // DisplayForm

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This is a community wiki answer. Feel free to improve it. Introduction RawArray is an atomic array type that can hold data in any of the following formats: "Integer8", "UnsignedInteger8", "Integer16", "UnsignedInteger16", "Integer32", "UnsignedInteger32", "Integer64", "UnsignedInteger64", "Real32", "Real64", "Complex64", "Complex128" (Some aliases can ...

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Such functions set up a one-time iterator, i.e. a GeneralUtilitiesIterator. Its functionality may be inspected with ? GeneralUtilitiesIterator and so forth. Except for a ConstantIterator which always returns the same value (forever), they go sequentially through values until they return GeneralUtilitiesIteratorExhausted. The main (if internal) utility ...

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27

So I have a little bit more info, but mostly a learning technique we can test: Init I wanted to see when each of these was called so I made each possible handler (without a previous assignment) just call Print. Then I found that for "Wolfram.System.Print" this is a terrible idea as it is called anytime the system prints. So I removed that. Here's what I'm ...

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CompileInnerDo This is the one that initially struck me as interesting since I use compiled functions quite a lot. From the documentation of Do: Unless an explicit Return is used, the value returned by Do is Null. But that doesn't seem to be the case for CompileInnerDo! f1 = Compile[{{x}}, Module[{a}, a = x; CompileInnerDo[a++, {i, 10^8}]] ...

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Properties for SparseArray and InterpolatingFunction objects SparseArray objects can accept a range of Properties (or Methods) that allow the efficient extraction of certain information, most commonly "AdjacencyLists" or "NonzeroPositions" as a frequently faster alternative to Position. I started this answer to detail them but as it grew I came to believe ...

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We can get old-style messages using Internal$MessageMenu = False Needless to say, this is unsupported and undocumented functionality. 25 Version 10 introduced some undocumented functionality that makes it easier to explore the style hierarchy. As usual, "undocumented" === "buyer beware". We start by evaluating: StyleManagerOpenStyleManager This does nothing other than trigger the auto-loading of certain symbols. We'll come back to a more interesting use of this symbol later. Now, we ... 25 StringToDouble was renamed StringToMReal as part of code cleanup and the addition of several friends: ?InternalStringTo* (*InternalStringToBoolean InternalStringToMInteger InternalStringToMReal InternalStringToMRational*) It should be as fast, but now in failure cases it returns a proper Failure object instead of some bizarro-$Failed[_String]...

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We can use one of several ways. First, let us create some test data: arr = Range[10]; (* this is an Integer packed array *) unpacked = {1, 2, 3} (* this is an Integer array that is NOT packed *) Since Mathematica 10.4, InternalPackedArrayType directly returns the type: InternalPackedArrayType[arr] (* Integer *) InternalPackedArrayType[unpacked] (* $... 22 One can use Style and FractionBoxOptions to set Beveled: Plot[Sin[x], {x, -2 Pi, 2 Pi}, Ticks -> {Pi Range[-2, 2, 1/2], Automatic}, BaseStyle -> {FractionBoxOptions -> {Beveled -> True}}] It doesn't look too good here, though: This looks ok, Style[3 Pi/2, FractionBoxOptions -> {Beveled -> True}] but not in the "Label" style, Style[... 22 A long comment: Automatic renaming again: ClearAll[F]; F[x_] := Hold[With[{a = x}, {b = x}, a]]; F[0] Hold[With[{a$ = 0}, {b = 0}, a]] I was expecting it already in DownValues, but this is not the case. DownValues @ F {HoldPattern[F[x_]] :> Hold[With[{a = x}, {b = x}, a]]} So who does it? :) Here is a pure guess, if evaluation is really a ...

21

In addition to the error messages quoted in the question the line returns: GeneralUtilitiesBenchmarkingPackagePrivateplot[ IndexBy[{{{16, 9.37132*10^-6}, . . . IndexBy was removed from 10.1.0: Note that IndexBy will be removed in a future version of Mathematica. It was something that was considered for 10.0.0 but didn't make the cut. – Stefan R Feb ...

21

Intro One day I was playing with Developer package and found DateSetter which uses some kind of floating elements that I found useful. After taking a look at a source code I found out it was FrontEndAttachedCell. From the code one could learn enough to create something useful: Usage FrontEndAttachCell[ parentObject, (*Box or Cell ...

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SystemPrivate*Entry* functions Since version 10.0 SystemPrivateSetNoEntry and SystemPrivateEntryQ functions are available. SystemPrivateSetNoEntry Changes internal representation of given expression so that it is considered atomic, it returns "atomized" version of expression. This change affects all references to given expression, but not new ...

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No, there are no more built-in generators. In this case, the SeedRandom::nogen message can be regarded as definitive. The list it prints is not predefined, but a runtime enumeration of all existing system random number generators. A couple of these are indeed undocumented: "Rule50025CA" is cellular automaton based and similar to "ExtendedCA" but using a ...

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In my view, the Mathematica front end would have needed a complete rewrite a long time ago to be in a shape where it can compete with modern UI's. Looking at MATLAB, it's clear that you always find things that are even more out-of-date. Also, I don't want to say that you can't create beautiful content inside a notebook. However, especially the capabilities ...

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“It wasn’t long before carriage makers were driving horseless carriages. It wasn’t long before people crossing the continent on trains abandoned the railroads for airliners. Natural gas is replacing coal and there is nothing the railroads, the coal miners, or the coal companies can do about it. Cheaper and more efficient energy always wins out over more expensive energy. Coal replaced wood, and oil replaced coal as the primary source of energy. Anything that is more efficient boosts the figures on the bottom line of the ledger. Dollars chase efficiency. Inefficiency is suppressed by market forces. Free Power wins in the market place.

Or, you could say, “That’s Free Power positive Delta G. “That’s not going to be spontaneous. ” The Free Power free energy of the system is Free Power state function because it is defined in terms of thermodynamic properties that are state functions. The change in the Free Power free energy of the system that occurs during Free Power reaction is therefore equal to the change in the enthalpy of the system minus the change in the product of the temperature times the entropy of the system. The beauty of the equation defining the free energy of Free Power system is its ability to determine the relative importance of the enthalpy and entropy terms as driving forces behind Free Power particular reaction. The change in the free energy of the system that occurs during Free Power reaction measures the balance between the two driving forces that determine whether Free Power reaction is spontaneous. As we have seen, the enthalpy and entropy terms have different sign conventions. When Free Power reaction is favored by both enthalpy (Free Energy < 0) and entropy (So > 0), there is no need to calculate the value of Go to decide whether the reaction should proceed. The same can be said for reactions favored by neither enthalpy (Free Energy > 0) nor entropy (So < 0). Free energy calculations become important for reactions favored by only one of these factors. Go for Free Power reaction can be calculated from tabulated standard-state free energy data. Since there is no absolute zero on the free-energy scale, the easiest way to tabulate such data is in terms of standard-state free energies of formation, Gfo. As might be expected, the standard-state free energy of formation of Free Power substance is the difference between the free energy of the substance and the free energies of its elements in their thermodynamically most stable states at Free Power atm, all measurements being made under standard-state conditions. The sign of Go tells us the direction in which the reaction has to shift to come to equilibrium. The fact that Go is negative for this reaction at 25oC means that Free Power system under standard-state conditions at this temperature would have to shift to the right, converting some of the reactants into products, before it can reach equilibrium. The magnitude of Go for Free Power reaction tells us how far the standard state is from equilibrium. The larger the value of Go, the further the reaction has to go to get to from the standard-state conditions to equilibrium. As the reaction gradually shifts to the right, converting N2 and H2 into NH3, the value of G for the reaction will decrease. If we could find some way to harness the tendency of this reaction to come to equilibrium, we could get the reaction to do work. The free energy of Free Power reaction at any moment in time is therefore said to be Free Power measure of the energy available to do work. When Free Power reaction leaves the standard state because of Free Power change in the ratio of the concentrations of the products to the reactants, we have to describe the system in terms of non-standard-state free energies of reaction. The difference between Go and G for Free Power reaction is important. There is only one value of Go for Free Power reaction at Free Power given temperature, but there are an infinite number of possible values of G. Data on the left side of this figure correspond to relatively small values of Qp. They therefore describe systems in which there is far more reactant than product. The sign of G for these systems is negative and the magnitude of G is large. The system is therefore relatively far from equilibrium and the reaction must shift to the right to reach equilibrium. Data on the far right side of this figure describe systems in which there is more product than reactant. The sign of G is now positive and the magnitude of G is moderately large. The sign of G tells us that the reaction would have to shift to the left to reach equilibrium.

I have had many as time went by get weak. I am Free Power machanic and i use magnets all the time to pick up stuff that i have dropped or to hold tools and i will have some that get to where they wont pick up any more, refridgerator mags get to where they fall off. Dc motors after time get so they don’t run as fast as they used to. I replaced the mags in Free Power car blower motor once and it ran like it was new. now i do not know about the neo’s but i know that mags do lose there power. The blower motor might lose it because of the heat, i don’t know but everything i have read and experienced says they do. So whats up with that? Hey Free Electricity, ok, i agree with what you are saying. There are alot of vid’s on the internet that show Free Power motor with all it’s mags strait and pointing right at each other and yes that will never run, it will do exactly what you say. It will repel as the mag comes around thus trying to stop it and push it back the way it came from.

Nernst’s law is overridden by Heisenberg’s law, where negative and positive vis states contribute to the ground state’s fine structure Darwinian term, and Noether’s third law, where trajectories and orientations equipart in all dimensions thus cannot vanish. Hi Paulin. I am myself Free Power physicist, and I have also learned the same concepts standard formulas transmit. However, Free Electricity points are relevant. Free Power. The equations on physics and the concepts one can extract from them are aimed to describe how the universe works and are dependent on empirical evidence, not the other way around. Thinking that equations and the concepts behind dogmatically rule empirical phenomena is falling into pre-illustrative times. Free Electricity. Particle and quantum physics have actually gotten results that break classical thermodynamics law of conservation of energy. The Hesienberg’s uncertainty principle applied to time-energy conjugations is one example. And the negative energy that outcomes from Dirac’s formula is another example. Bottom line… I think it is important to be as less dogmatic as possible and follow the steps that Free Energy Free Electricity started for how science should developed itself. My Name is Free Energy Sr and i have made Free Power Magnetic motor.

Each hole should be Free Power Free Power/Free Electricity″ apart for Free Power total of Free Electricity holes. Next will be setting the magnets in the holes. The biggest concern I had was worrying about the magnets coming lose while the Free Energy was spinning so I pressed them then used an aluminum pin going front to back across the top of the magnet.

The net forces in Free Power magnetic motor are zero. There rotation under its own power is impossible. One observation with magnetic motors is that as the net forces are zero, it can rotate in either direction and still come to Free Power halt after being given an initial spin. I assume Free Energy thinks it Free Energy Free Electricity already. “Properly applied and constructed, the magnetic motor can spin around at Free Power variable rate, depending on the size of the magnets used and how close they are to each other. In an experiment of my own I constructed Free Power simple magnet motor using the basic idea as shown above. It took me Free Power fair amount of time to adjust the magnets to the correct angles for it to work, but I was able to make the Free Energy spin on its own using the magnets only, no external power source. ” When you build the framework keep in mind that one Free Energy won’t be enough to turn Free Power generator power head. You’ll need to add more wheels for that. If you do, keep them spaced Free Electricity″ or so apart. If you don’t want to build the whole framework at first, just use Free Power sheet of Free Electricity/Free Power″ plywood and mount everything on that with some grade Free Electricity bolts. That will allow you to do some testing.
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.
Your Free Power typical narrow-minded democrat. They are all liars, cowards, cheats and thieves. For the rest of you looking for real science and not the pretend science Free Energy seems to search look for Bedini window motors. Those seem to be the route to generating 5kw for your house. Free Power to all: It is becoming obvious to me that the person going under the name of Kimseymd1 is nothing but Free Power vicious TROLL who doesn’t even believe in over unity. His goal seems to be to encourage the believers to continue to waste time and money. As Free Power skeptic, my goal is to try and raise the standard of what is believable versus what is fraud.
The net forces in Free Power magnetic motor are zero. There rotation under its own power is impossible. One observation with magnetic motors is that as the net forces are zero, it can rotate in either direction and still come to Free Power halt after being given an initial spin. I assume Free Energy thinks it Free Energy Free Electricity already. “Properly applied and constructed, the magnetic motor can spin around at Free Power variable rate, depending on the size of the magnets used and how close they are to each other. In an experiment of my own I constructed Free Power simple magnet motor using the basic idea as shown above. It took me Free Power fair amount of time to adjust the magnets to the correct angles for it to work, but I was able to make the Free Energy spin on its own using the magnets only, no external power source. ” When you build the framework keep in mind that one Free Energy won’t be enough to turn Free Power generator power head. You’ll need to add more wheels for that. If you do, keep them spaced Free Electricity″ or so apart. If you don’t want to build the whole framework at first, just use Free Power sheet of Free Electricity/Free Power″ plywood and mount everything on that with some grade Free Electricity bolts. That will allow you to do some testing.
If power flows from the output shaft where does it flow in? Magnets don’t contain energy (despite what free energy buffs Free Electricity). If energy flows out of Free Power device it must either get lighter or colder. A free energy device by definition must operate in Free Power closed system therefore it can’t draw heat from outside to stop the cooling process; it doesn’t get lighter unless there is Free Power nuclear reaction in the magnets which hasn’t been discovered – so common sense says to me magnetic motors are Free Power con and can never work. Science is not wrong. It is not Free Power single entity. Free Electricity or findings can be wrong. Errors or corrections occur at the individual level. Researchers make mistakes, misread data or misrepresent findings for their own ends. Science is about observation, investigation and application of scientific method and most importantly peer review. Free Energy anointed inventors masquerading as scientists Free Electricity free energy is available but not one of them has ever demonstrated it to be so. Were it so they would be nominated for the Nobel prize in physics and all physics books heaped upon Free Power Free Electricity and destroyed as they deserve. But this isn’t going to happen. Always try to remember.
The internet is the only reason large corps. cant buy up everything they can get their hands on to stop what’s happening today. @Free Power E. Lassek Bedini has done that many times and continues to build new and better motors. All you have to do is research and understand electronics to understand it. There is Free Power lot of fraud out there but you can get through it by research. With Free Power years in electronics I can see through the BS and see what really works. Build the SG and see for yourself the working model.. A audio transformer Free Power:Free Power ratio has enough windings. Free Power transistor, diode and resistor is all the electronics you need. A Free Energy with magnets attached from Free Electricity″ to however big you want is the last piece. What? Maybe Free Electricity pieces all together? Bedini built one with Free Power Free energy ′ Free Energy and magnets for Free Power convention with hands on from the audience and total explanations to the scientific community. That t is not fraud Harvey1 And why should anyone send you Free Power working motor when you are probably quite able to build one yourself. Or maybe not? Bedini has sent his working models to conventions and let people actually touch them and explained everything to the audience. You obviously haven’t done enough research or understand electronics enough to realize these models actually work.. The SC motor generator is easily duplicated. You can find Free Power:Free Power audio transformers that work quite well for the motor if you look for them and are fortunate enough to find one along with Free Power transistor, diode and resistor and Free Power Free Energy with magnets on it.. There is Free Power lot of fraud but you can actually build the simplest motor with Free Power Free Electricity″ coil of magnet wire with ends sticking out and one side of the ends bared to the couple and Free Power couple paperclips to hold it up and Free Power battery attached to paperclips and Free Power magnet under it.
Free Power you? Im going to stick to the mag motor for now. Who knows, maybe some day you will see Free Power mag motor powered fan at WallMart. Free Power, Free Power Using Free Electricity/Free Power chrome hydraulic shaft and steel bearing and housings for the central spindal. Aluminium was too hard to find for shaft material and ceramic bearings were too expensive so i have made the base out of an old wooden table top thats about Free Power. 3metres across to get some distance. Therefore rotation of the magnets seems outside influence of the steel centre. Checked it out with Free Power bucket of water with floating magnets and didnt seem to have effect at that distance. Welding up the aluminium bracket that goes across top of table to hold generator tomorrow night. Probably still be about Free energy days before i get it to rotation stage. Looks awesome with all the metal bits polished up. Also, I just wanted to add this note. I am not sure what to expect from the design. I am not claiming that i will definitely get over unity. I am just interested to see if it comes within Free Power mile of it. Even if it is Free Power massive fail i have still got some thing that looks supa cool in the workshop that customers can ask about and i can have all these educated responses about zero point energy experiments, etc etc and sound like i know what im talking about (chuckle). After all, having Free Power bit of fun is the main goal. Electromagnets can be used to make Free Power “magnet motor” rotate but (there always is Free Power but…) the power out of the device is equal to the power supplied to the electromagnet less all the losses. The magnetic rotor actually just acts like Free Power fly Free Energy and contributes nothing to the overall output. Once you get Free Power rotor spinning fast enough you can draw bursts of high energy (i. e. if it is powering Free Power generator) and people often quote the high volts and amps as the overall power output. Yippee OVERUNITY! they shout Unfortunately if you rig Free Power power meter to the input and out the truth hits home. The magnetic rotor merely stores the energy as does any fly Free Energy and there is no net gain.
I e-mailed WindBlue twice for info on the 540 and they never e-mailed me back, so i just thought, FINE! To heck with ya. Ill build my own. Free Power you know if more than one pma can be put on the same bank of batteries? Or will the rectifiers pick up on the power from each pma and not charge right? I know that is the way it is with car alt’s. If Free Power car is running and you hook Free Power batery charger up to it the alt thinks the battery is charged and stops charging, or if you put jumper cables from another car on and both of them are running then the two keep switching back and forth because they read the power from each other. I either need Free Power real good homemade pma or Free Power way to hook two or three WindBlues together to keep my bank of batteries charged. Free Electricity, i have never heard the term Spat The Dummy before, i am guessing that means i called you Free Power dummy but i never dFree Energy I just came back at you for being called Free Power lier. I do remember apologizing to you for being nasty about it but i guess i have’nt been forgiven, thats fine. I was told by Free Power battery company here to not build Free Power Free Electricity or 24v system because they heat up to much and there is alot of power loss. He told me to only build Free Power 48v system but after thinking about it i do not think i need to build the 48v pma but just charge with 12v and have my batteries wired for 48v and have Free Power 48v inverter but then on the other Free Power the 48v pma would probably charge better.

“It wasn’t long before carriage makers were driving horseless carriages. It wasn’t long before people crossing the continent on trains abandoned the railroads for airliners. Natural gas is replacing coal and there is nothing the railroads, the coal miners, or the coal companies can do about it. Cheaper and more efficient energy always wins out over more expensive energy. Coal replaced wood, and oil replaced coal as the primary source of energy. Anything that is more efficient boosts the figures on the bottom line of the ledger. Dollars chase efficiency. Inefficiency is suppressed by market forces. Free Power wins in the market place.
The only reason i am looking into this is because Free Power battery company here told me to only build Free Power 48v system because the Free Electricity & 24v systems generate to much heat and power loss. Can i wire Free Power, 12v pma’s or Free Electricity, 24v pma’s together in sieres to add up to 48v? If so i do not know how to do it and will that take care of the heat problem? I am about to just forget it and just build Free Power 12v system. Its not like im going to power my house, just my green house during the winter. Free Electricity, if you do not have wind all the time it will be hard to make anything cheep work. Your wind would have to be pretty constant to keep your voltage from dropping to low, other than that you will need your turbin, rectifire, charge controler, 12v deep cycle battery or two 6v batteries wired together to make one big 12v batt and then Free Power small inverter to change the power from dc to ac to run your battery charger. Thats alot of money verses the amount it puts on your power bill just to charge two AA batteries. Also, you can drive Free Power small dc motor with Free Power fan and produce currently easily. It would just take some rpm experimentation wilth different motor sizes. Kids toys and old VHS video recorders have heaps of dc motors.

Vacuums generally are thought to be voids, but Hendrik Casimir believed these pockets of nothing do indeed contain fluctuations of electromagnetic waves. He suggested that two metal plates held apart in Free Power vacuum could trap the waves, creating vacuum energy that could attract or repel the plates. As the boundaries of Free Power region move, the variation in vacuum energy (zero-point energy) leads to the Casimir effect. Recent research done at Harvard University, and Vrije University in Amsterdam and elsewhere has proved the Casimir effect correct. (source)

Why not use the term over unity over perpetual motion? Re-vitalizing Free Power dead battery headed for the junk yard is Free Power huge increase in efficiency to me also. Why doesn’t every AutoZone or every auto shop have one of these? Unless the battery case is cracked every battery could be reused. The charge of Free Power re-vitalize instead of Free Power new battery. Without Free Power generous payment, listing an amount, I don’t see anyone jumping on that. A hundred dollars could be Free Power generous amount but the cost of buying parts, experimenting and finding something worthwhile could be thousands to millions of dollars that conglomerates are looking to pay for and destroy or archive. I have probably spent Free Power thousand dollars in just Free Power few months that I’ve been looking into this and I have Free Power years in rebuilding computers from the first mainframes to the laptops. I retired and now its Free Power hobby. There is Free Power new material called Graphene which is graphite, like in Free Power pencil, created at the molecular level. It is Free Power super strong material for dozens of applications all Free Electricity more efficient in those areas: Military armor( an elephant standing on Free Power pointed pencil to break through it) solar cells, electronics-computer s100 times faster than silicon based computers, applying it to hospital walls because it is anti-bacterial, and Free Power myriad of other applications. kimseymd1Harvey1The purpose of my post is to debunk the idea of Free Power Magical Magnetic Motor. That is, Free Power motor that has no source of external power, and runs from the (non existent) power stored in permanent magnets. Advances made to electric motors in the past few years are truly amazing, but are totally outside the scope of my post.

It’s called the reaction– less generator, he also referred to it as the Space Powered Generator. It allows for the production of power with improved efficiency. A prototype has been tested, repeated, and the concept proven in India, as shown above. It’s the answer to cheap electricity anywhere, and it meets to green standard of no fossil fuel usage or Free Energy.
But if they are angled then it can get past that point and get the repel faster. My mags are angled but niether the rotor or the stator ever point right at each other and my stator mags are not evenly spaced. Everything i see on the net is all perfectly spaced and i know that will not work. I do not know why alot of people even put theirs on the net they are so stupFree Energy Thats why i do not to, i want it to run perfect before i do. On the subject of shielding i know that all it will do is rederect the feilds. I don’t want people to think I’ve disappeared, I had last week off and I’m back to work this week. I’m stealing Free Power little time during my break to post this. Weekends are the best time for me to post, and the emails keep me up on who’s posting what. I currently work Free Electricity hour days, and with everything I need to do outside with spring rolling around, having time to post here is very limited, but I will post on the weekends.
But I will send you the plan for it whenever you are ready. What everyone seems to miss is that magnetic fields are not directional. Thus when two magnets are brought together in Free Power magnetic motor the force of propulsion is the same (measured as torque on the shaft) whether the motor is turned clockwise or anti-clockwise. Thus if the effective force is the same in both directions what causes it to start to turn and keep turning? (Hint – nothing!) Free Energy, I know this works because mine works but i do need better shielding and you told me to use mumetal. What is this and where do you get it from? Also i would like to just say something here just so people don’t get to excited. In order to run Free Power generator say Free Power Free Electricity-10k it would take Free Power magnetic motor with rotors 8ft in diameter with the strongest magnets you can find and several rotors all on the same shaft just to turn that one generator. Thats alot of money in magnets. One example of the power it takes is this.
This expression has commonly been interpreted to mean that work is extracted from the internal energy U while TS represents energy not available to perform work. However, this is incorrect. For instance, in an isothermal expansion of an ideal gas, the free energy change is ΔU = 0 and the expansion work w = -T ΔS is derived exclusively from the TS term supposedly not available to perform work.


text:
Equivalence of weighted Minkowski sums

Given $n$ polytopes $P_1, \cdots, P_n$, each $P_i$ is given as the V-representation, i.e., a set of $m$ points as its set of vertices.

Furthermore, consider a variant of the Minkowski sum (somehow its weighted version). For $n$ nonnegative numbers $\vec{w}=(w_1, \cdots, w_n)$, we write $\sum_{\vec{w}}P_i=\{\sum_{i=1}^n w_i \vec{x}_i\mid \vec{x}_i\in P_i\}$.

The question is, given two weight vectors $\vec{w}$ and $\vec{v}$, decide whether $\sum_{\vec{w}}P_i = \sum_{\vec{v}}P_i$, where $=$ is interpreted as set equivalence.

It appears that a straightforward approach fails because $\sum_{\vec{w}}P_i$ as a polytope, might contain exponentially many vertices (?, order of $m^n$), so I only can show that the problem is in coNP. But is the problem in P, or it is coNP-hard as well?


text:
# Probability of number of events where $p_{n+1} = \frac{1}{5}p_n$

For $n \ge 0, p_{n+1} = \frac{1}{5}p_n$. Where $p_n$ is the probability of an envent occuring n times in a period.

What is the probability that more than one event occurs?

-

$1= \sum\limits_{n=0}^{\infty}(\frac{1}{5})^{n}p_0$

$(\frac{1}{5})(1) = (\frac{1}{5})\sum\limits_{n=0}^{\infty}(\frac{1}{5})^{n}p_0$
$1 - (\frac{1}{5})(1) = p_0 (\frac{1}{5})^{0}$
$\frac{4}{5} = p_0$

using the formula in the question,
$p_1 = \frac{1}{5}(\frac{4}{5})$
$p_1 = \frac{4}{25}$

$P(n\ge 2) = 1 - P(n=0) - P(n=1)$
$P(n\ge 2) = 1 - \frac{4}{5} - \frac{4}{25}$
$P(n\ge 2) = 0.04$

-

The probability of more than one event is $1-p_0-p_1$, which is $1-p_0-\frac{p_0}{5}$.

Note that $p_k=p_0\left(\frac{1}{5}\right)^k$. But $p_0+p_1+p_2+\cdots=1$. Thus $$p_0\left(1+\frac{1}{5}+\left(\frac{1}{5}\right)^2+\left(\frac{1}{5}\right)^3+ \cdots\right)=1.$$ Summing the geometric series, we find that $p_0(5/4)=1$.

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text:
Introduction

The Sun sporadically produces eruptive events, such as flares and coronal mass ejection, that can lead to highly intense fluxes of solar energetic particles (SEPs), which escape into the interplanetary space and possibly hit Earth1. When reaching the Earth, SEPs can have a dramatic impact on modern communication, navigation and power systems, satellite life expectancy, the health of astronauts, and aircraft operations2,3. The famous Carrington event was one of the strongest known solar storms that hit the Earth in September 1859 CE, leading to a widespread failure of the telegraph system and observable auroras all over the world3,4. If this, or an even larger event, occurred today the impact on global society and economy would be catastrophic3,5.

The first instrumental observations of SEPs were conducted in the 1940s6. Since then their physical origin, frequency of occurrence, amplitude, and energy distribution have been studied using ground-based and space-borne data7. The strongest measured SEP event (called ground-level-enhancement GLE #5) took place on 23-Feb-1956 (a list of directly measured events is available at the International GLE database (https://gle.oulu.fi)). However, little is known about extreme SEP events whose very existence was unknown until a few years ago8. Even though observational advances such as a worldwide network of neutron monitors established in 1951/1957/1964 made it possible to characterize the Suns’ eruptive behavior9, the temporal coverage of the observational record is not long enough to assess the frequency of extremely rare but highly energetic SEP events. Statistics of sun-like stars suggest that the superflares that produce highly energetic SEPs are extremely rare10,11.

Understanding and ultimately predicting extreme solar events will not only help in mitigating their harmful consequences on modern life and communication systems, but will also help understanding the complex magneto-hydrodynamic behavior of the Sun. In this context, cosmogenic radionuclides provide a powerful tool for reconstructing past solar activity and SEP events12,13,14,15.

Cosmogenic radionuclides such as 10Be, 14C and 36Cl are mainly produced by galactic cosmic rays (GCRs) originating from outside our solar system, hitting the Earth’s atmosphere. The flux of GCRs on Earth is modulated (shielded) by the geomagnetic field and the solar magnetic field carried by the solar wind16. As a consequence, enhanced magnetic fields (solar/heliospheric or geomagnetic) reduce cosmogenic nuclide production by GCRs on Earth and vice versa. When hitting the Earth, strong SEPs may cause an additional, short-term increase in cosmogenic nuclide production. Natural archives, such as dendrochronologically dated tree-rings (14C) or polar ice cores (10Be,36Cl), are known to provide precise and temporally accurate records of past cosmogenic radionuclide production12,17,18 giving us a unique opportunity to identify and study the characteristics of strong SEP events over the past thousands of years.

So far, three strong SEP events that led to an abrupt increase of about 1% or more in atmospheric 14C concentrations within less than two years have been unambiguously detected over the past 3,000 years in tree-rings, and confirmed with other cosmogenic radionuclides (10Be, 36Cl) in ice cores, in the years 993 CE, 775 CE and 660 BCE13,14,19,20. More, weaker, yet unconfirmed SEP events have been found in the radionuclide records17,21,22 (Table 1). They are currently considered candidate events either because there is insufficient data coverage and precision to clearly distinguish them from normal solar modulation or because they have not yet been confirmed23. The above mentioned Carrington Event did not lead to a detectable increase in cosmogenic radionuclides17,24,25 implying that associated SEPs either missed the Earth or did not have sufficient energy.

The longest absolutely dated composite tree-ring record spans the last 12,460 years26. At present, high-precision, annual 14C measurements from tree-rings, which are required for the sensitive detection of strong SEP events, only cover about 2030 years27. Therefore, our ability to study strong SEP events is limited and theoretical estimations on their frequency9 and energy spectra20,28 remain highly uncertain. The extension of the time period covered with annually resolved 14C measurements in tree-rings will provide statistically robust estimates on the magnitude and frequency of strong SEP events.

Currently, the search for strong SEP events mainly relies on existing cosmogenic nuclide records with lower (5–20 years) temporal resolution. The motivation to investigate the annually resolved 14C signal in tree-rings between 7150 BCE and 7200 BCE came from lower resolution, synchronized29,30 10Be and 36Cl data in ice cores31,32,33 with a resolution of 5 to 20 years and from decadal 14C data (IntCal20)27. Another event in 5259 BCE, was discovered while investigating unexpected difficulties in producing stable Bayesian chronological models for archaeological sites and ceramic sequences dating to the period centering on the 53rd century BCE34,35,36. With hindsight, this event could not have been detected by the statistical analysis of cosmogenic nuclides from the lower resolution ice core data37.

Here we present two previously unknown 14C production events recorded in two (7176 BCE) and four (5259 BCE) independent, absolutely dated tree-ring chronologies. The magnitudes of the events are assessed by calculating the additional 14C production using a global carbon cycle box model. They are compared to other known 14C events by normalizing their magnitudes to modern geomagnetic shielding. Furthermore, the 7176 BCE event is used as a unique time marker to place a floating (i.e. not absolutely dated) portion of the Bristlecone Pine tree-ring chronology (USA) within 1–2 years on its absolutely dated part.

Results

Atmospheric 14C concentrations measured in several independently built tree-ring chronologies increased by nearly 2% within two years in 7176 BCE and 5259 BCE (Fig. 1, Table 1, Supplementary Fig. 1) thereby exceeding the rise in 14C concentration of the largest known event in 775 CE14. For the 7176 BCE event (Fig. 1a) an average increase of (19.5 ± 0.6)‰ in 14C concentrations was observed in the two existing absolutely dated tree-ring chronologies from central Europe covering that time range, the German oak and the Eastern Alpine conifer38,39 chronologies. The 7176 BCE event was also found in a currently floating portion of the Bristlecone Pine chronology from the USA (Supplementary information S1). The 5259 BCE event (Fig. 1b) was found in four absolutely dated tree-ring chronologies from the Alps, Ireland, Russia and the USA showing an average increase in atmospheric 14C concentrations of (19.1 ± 0.6) ‰. A potentially early increase is observed in the Bristlecone Pine in 5260 BCE.

Total and excess 14C productions from the SEP event were calculated using a global carbon cycle model17 (Methods). The excess production (and related uncertainty) was calculated using a Monte–Carlo approach by fitting offset and amplitude of a Gaussian-shaped production spike to the data (Fig. 2, Methods). For each of the 1000 different Monte–Carlo data realizations the amount of additional 14C produced was extracted. The average amount of 14C excess produced during the two events was (29.2 ± 0.9) kg for the 5259 BCE event and (28.7 ± 0.9) kg for the 7176 BCE event (Table 1). This is more than (7176 BCE) or comparable to (5259 BCE) the amount of 14C produced by the strongest 14C event detected so far (775 CE, (26.2 ± 1.0) kg excess 14C). In both cases, a modeled duration of 0.3 year, 2-sigma of the Gaussian, was used to fit the measured 14C increase. This is significantly less than the tree-ring resolution of one year.

Excess 14C production of all known events (Supplementary Fig. 3) was (re-)calculated using the above described procedure (Fig. 2a). When comparing different events, it is, however, important to note that the same event (in terms of the proton flux and energy distribution) can lead to different excess productions of cosmogenic radionuclides depending on the strength of the Earth’s geomagnetic field at the time of the event. A weak geomagnetic field would result in higher excess production of cosmogenic radionuclides by the SEP event and vice versa. We therefore recalculated the excess of 14C production by normalizing all events relative to the modern geomagnetic dipole moment of 7.8 ∙ 1022 Am2 using two different geomagnetic field records40,41 (see Methods/ Supplementary Figs. 4 and 5 (Fig. 3). The resulting normalized excess 14C productions for the 7176 BCE event are not significantly different from the original value (Table 1). The 5259 BCE event occurred during the period of weaker magnetic field strength and, thus, referring to modern geomagnetic shielding the excess 14C production would be slightly lower. The normalization procedure also significantly changes the excess 14C production of the other known events. For the 775 CE event, the two geomagnetic field reconstructions give different results, while the Knudsen41 reconstruction leads to a 20% increase of excess 14C, using the Panovska40 reconstruction increases it by only 10%. When comparing the events after normalization we observe that the 7176 BCE and the 5259 BCE events are of similar, but slightly weaker magnitude than the 775 CE event. In any case, they are significantly stronger than both 660 BCE and 993 CE events.

Discussion

The two identified events are assigned to the years 5259 BCE and 7176 BCE in several independent and absolutely dated tree-ring chronologies. The fact that the events are accurately dated to one year makes them (and all other 14C events) excellent time markers for the synchronization of chronologies. For example, an older floating portion of the USA Bristlecone chronology could not be comprehensively connected to its master chronology using tree-ring dating techniques, due to a very short (not statistically verifiable) period of overlap between the records. Instead, this chronology was assigned possible calendar dates based upon a tentative dendrochronological match within a broader time window suggested by conventional radiocarbon dating42 (see SI for full details). The results of this analysis clearly showed the 7176 BCE event, placing this previously floating sequence on an absolute timescale. Further substantiation of this connection will provide an extension of the Methuselah Walk bristlecone pine chronology beyond 6827 BCE (where the calendar dated portion ends) across a full 10,399 year sequence.

For the 5259 BCE event, we note that the bristlecone record shows a potential early increase in 5260 BCE. There are a number of possible causes for this. Very narrow ring widths (<0.5 mm) during this period, indicating difficult environmental conditions, may have hampered the complete dissection of the rings, or there may be a previously undetected dating error in this earlier calendar dated portion of the record, where cross-checking with other site chronologies is not possible. Alternatively, this anomalous result may relate to regional shifts in growing season or physiological differences between deciduous and indeciduate trees. If, for example, the 14C event occurred in late summer or autumn of 5260 BCE, i.e. after the end of the ring formation period in Ireland, the Alps and the Russian north, but before the end of tree ring formation in California, this would result in increased 14C content of the 5260 BCE bristlecone pine tree ring, but not in the other chronologies. However, this cannot be argued on the basis of published information on the end of today’s growing seasons (Bristlecone pine—late August43,44, Alpine larch—October45, Irish oak – October46, Siberian larch-late August). Furthermore, the fact that the event occurred during the Holocene climatic optimum, which was characterized by a weaker latitudinal temperature gradient47, may mean that more synchronous growing seasons would be more likely. Another possibility may relate to the fact that Bristlecone Pine is the only indeciduate tree in this record. Deciduous trees such as oak and larch store photosynthates produced during the end of growing season to grow the next year’s earlywood48. If the 14C event occurred towards the very end of the deciduous tree’s growth season it is possible that only the indeciduate Bristlecone Pine would register the change in the same year. If this early increase of 14C in Bristlecone Pine can be confirmed by future replicate measurements, the timing of the event would be confined precisely to the end of summer/autumn 5260 BCE.”

Polar ice cores, which are regarded as extremely valuable climate archives, have typically been dated with a precision of about 10 years within the Holocene29 and even less in the transition to the glacial period42. The absolutely dated 14C events in tree-rings can now be used to synchronize the ice core chronologies via cosmogenic 10Be and 36Cl resulting in a precision of about 1–2 years limited by several factors such as weather noise, sampling, and radionuclide transfer times to the ice core43. As a result, chronologies that so far were synchronized with absolutely dated tree-ring records using the low resolution structure of existing 14C or 10Be records29 can now be placed within 1–2 years or about 10 times more precisely around these events using the abrupt change in radionuclide concentrations caused by SEP events.

The increasing number of discoveries of strong SEP events that hit Earth over the past 12,000 years indicates that they cannot be considered as extremely rare anymore. So far, only a few sequences covering a total of 2030 yrs of the past 12,400 years (16.5%) have been analyzed with 14C at the annual or biennial resolution, which is required to unambiguously detect SEP events, revealing five strong SEP events (Fig. 4). These findings might lead to the conclusion that strong SEP events hit Earth once every about 400 years on average (i.e. five events in 2030 years). This simple statistic, however, is likely biased by the fact that some studied time periods were not randomly selected for annual analysis, but rather were targeted based on indications from lower-resolution cosmogenic datasets or archaeological evidence that the present multi-annual 14C record might contain more structures than previously visible. In an unlikely case that all major events have already been found over the last 12,400 years, the lower limit of occurrence for strong SEP events can be estimated as five events in 12,400 years or one every ~2400 years. Based on our findings we can constrain the occurrence rate of strong solar events to one every 400–2400 years. We nevertheless expect more events to be discovered as additional annually resolved data becomes available, leading to more precise estimates of the frequency of occurrence and magnitude of strong SEP events. The estimate of the occurrence rate of strong SEP events shows that they are more frequent than suggested by the revised statistics of superflares on sun-like stars (once every 3000--6000 yr)11. However, a direct relation between superflares and strong SEP events remains unknown. The statistical constraints provided by the data will help to test different approaches in solar/stellar physics describing the occurrence and magnitude of extreme events in sun-like stars. Ranking among the three largest short-term 14C production events, the impact of the newly discovered events would have been catastrophic for aircraft, satellites, modern telecommunication and computer systems44,45,46, if they occurred today.

Materials and Methods

Sample preparation and measurement

For the 7176 BCE event dendrochronologically dated wood samples from Ireland, supplied by the University of Belfast, and from the Alps, supplied by the University of Innsbruck, were dissected into annually resolved samples weighing 30–60 mg.

For the analyses performed at ETHZ, typically 54 tree-ring samples with four wood blanks (2 BC and 2 KB) and 2 1515 CE reference samples47 each weighing 30–60 mg, were prepared in 15 ml glass test tubes together in a batch (making 60 in total). In a slightly modified procedure following Němec et al.48, samples were first soaked in 5 ml 1 M NaOH overnight at 70 °C in an oven. Then the samples were treated with 1 M HCl and 1 M NaOH for 1 hour each at 70°C in a heat block, before they were bleached at a pH of 2–3 with 0.35 M NaClO4 at 70 °C for 2 h. The remaining white holo-cellulose was then freeze-dried overnight.

About 2.5 mg dried holo-cellulose was wrapped in cleaned Al capsules49 and converted to graphite using the automated graphitization line AGE50. A measurement set was made up of three oxalic acid one (OX1) and four oxalic acid two (OX2) standards, 27 samples, two cellulose blanks, two chemical blanks, and two 1515 CE reference samples (individual cellulose preparations of the 1515 CE reference is used for at least two measurements) and measured in the MICADAS accelerator mass spectrometer51. Two measurement sets were typically prepared from one set of samples within a week and subsequently measured. A second graphite sample was subsequently prepared and measured from one third of the prepared cellulose samples for quality control purposes.

Two internal wood reference materials from 1515 CE (Pine and Oak) and two different radiocarbon free wood blanks (Kauri Stage 7, KB, and Brown Coal, BC, from Reichwalde) were repetitively analyzed together with the annual samples. While the wood-blanks were used for blank subtraction in the data evaluation process, the 1515 CE references were used for quality control only.

For the analyses performed in Bristol, tree ring samples (20–30 mg) were processed alongside two KB wood blanks and two oak 1524 CE reference samples in 15 ml glass culture tubes. Samples were pretreated following the standard BRAMS wood pretreatment method (code BABAB; following Němec et al.), as described by Knowles et al.52. Around 2.5 mg dried holo-cellulose was combusted, graphitized, and measured using the Bris-MICADAS AMS. Full details of the procedures employed are described by Knowles et al.52.

Several samples were repetitively measured, and a chi-squared analysis showed that the repeated measurements were generally in good agreement with one another. The resulting chi-squared values of the different chronologies are shown in Table 2. Small observed differences are likely due to wood anatomical effects or due to the limitations of cutting tree-rings accurately for species with narrow tree-rings (Supplements S1). The high overall χ2 value events are likely due to regional offset53 and the earlier increase signal in the Bristlecone Pine chronology in the 5259 BCE event.

Modeling

To model the carbon cycle an improved carbon box model based on the model of Güttler et al.54 was used (Supplementary Fig 2). The model of Güttler uses 11 Boxes to simulate the exchange between the global atmosphere biosphere and Oceans. To model the northern and southern hemispheres separately our model was extended to 22 boxes (11 boxes for each hemisphere). The carbon content of each box is distributed according to the respective relative carbon reservoir masses of the corresponding hemisphere. Radiocarbon is produced in the stratosphere and the troposphere of both hemispheres, where 70% is produced in the Stratosphere and 30 % in the Troposphere. The fluxes were adjusted to ensure a correct ∆14C offset between the northern and southern troposphere. Seasonal variability of fluxes was not considered in the model.

The 12C and 14C content of each box after a time step is calculated with

$${N}_{i}^{12}(t+\Delta t)={N}_{i}^{12}(t)+d{N}_{i}^{12}(t)\Delta t$$
(1)
$${N}_{i}^{14}(t+\Delta t)={N}_{i}^{14}(t)+d{N}_{i}^{14}(t)\Delta t$$
(2)
$$d{N}_{i}^{12}(t)=\mathop{\sum}\limits_{j}{F}_{ji}^{12}(t)-\mathop{\sum}\limits_{j}{F}_{ij}^{12}(t)$$
(3)
$$d{N}_{i}^{14}(t)=-\lambda {N}_{i}+\mathop{\sum}\limits_{j}{F}_{ji}^{14}(t)-\mathop{\sum}\limits_{j}{F}_{ij}^{14}(t)+{P}_{i}(t)+{P}_{st,i}$$
(4)

Here $${N}_{i}^{12,14}$$ is the $${C}^{12,14}$$ content of each box in Gt and λ is the decay constant of 14C. The time step ∆t was chosen to be one month for all the following simulations. The 14C and 12C fluxes are given by the following:

$${F}_{ij}^{12}(t)={F}_{ij,st}\frac{{N}_{i}^{12}(t)}{{N}_{i,st}^{12}},{F}_{ij}^{14}(t)={F}_{ij,st}\frac{{N}_{i}^{12}(t)}{{N}_{i,st}^{12}}\frac{{N}_{i}^{14}(t)}{{N}_{i}^{12}(t)}\frac{{m}_{14}}{{m}_{12}}={F}_{ij,st}\frac{{N}_{i}^{14}(t)}{{N}_{i,st}^{12}}\frac{{m}_{14}}{{m}_{12}}$$
(5)

Here $${F}_{ij,st}$$ are the fluxes given in Supplementary Fig. 2. The fluxes are scaled depending on the deviation from the Holocene steady state $${N}_{i,st}^{12}$$. The steady state was computed by simulating 200’000 years with the constant production rate $${p}_{st}=1.76\frac{at}{c{m}^{2}s}$$. The model does not consider any isotopic fractionation and thus the 14C fluxes scale just as the 12C fluxes.

With this a general expression for all boxes at any time can be achieved.

$$N(t+\Delta t)=N(t)+(\varLambda N(t)+{F}^{T}(t)1-F(t)1+P(t)+{P}_{st})\Delta t$$
(6)
$$F =\left[\begin{array}{cc}{F}^{12}(t) & 0\\ 0 & {F}^{14}(t)\end{array}\right],\varLambda =\left[\begin{array}{cc}0 & 0\\ 0 & -\lambda \end{array}\right],N(t)=\left(\begin{array}{c}{N}^{12}\\ {N}^{14}\end{array}\right),{1}_{i}=1\;{{{\mbox{forall}}}}\;i\\ \,P(t) =Vp(t),{P}_{st}=V{p}_{st}$$
(7)
$${{V}_{i}}=\left\{\begin{array}{cc}0.5\cdot 0.7,\,if\,i=Stratosphere\,N,S\\ 0.5\cdot 0.3,\,if\;i=Troposphere \,N,S \\ 0,\hfill else\hfill\end{array}\right.$$
(8)

The ∆14C of each box for the simulation is calculated by the following expression:

$${\Delta }^{14}{C}_{i}(t)=\frac{\frac{{N}_{i}^{14}(t)}{{N}_{i}^{12}(t)}-\frac{{N}_{TN,st}^{14}}{{N}_{TN,st}^{12}}}{\frac{{N}_{TN,st}^{14}}{{N}_{TN,st}^{12}}}\cdot 1000$$
(9)

Where $${N}_{TN,st}^{12,14}$$ are the steady state 12,14C contents of the northern troposphere.

To get a reasonable start state at a given time the production rate for the whole IntCal13 record has been calculated and the simulated state can be loaded at any time before 1950 CE.

The events were evaluated by generating 1000 realizations of the data distributed with the errors of the measurements. For each realization a Gaussian shaped production spike was fitted and the additional production was extracted by integrating the Gaussian production. Since the energy spectrum of the SEPs is softer than the one of the galactic cosmic rays the excess production of the event was mainly produced in the stratosphere (90%) and only 10% in the troposphere. The errors were estimated by using the widths of the resulting distributions.

Geomagnetic field correction

Since the production of 14C in the Earth’s atmosphere is affected by geomagnetic shielding of the flux of SEPs, this effect needs to be accounted for in such a way that all the events are normalized to the same reference standard, specifically to the modern conditions with the dipole moment M = 7.8·1022 A·m2 55. The geomagnetic field has a very complex structure at the Earth’s surface56, but for energetic particles, the dipole moment is the most important since higher momenta decay faster with distance, and the eccentric+tilted dipole approximation is widely used57. Since the true dipole moment is difficult to determine in the past, before the era of direct measurements, an approximation of the virtual axial dipole moment (VADM) is reconstructed from paleo- and archeo-magnetic models58. Accordingly, the geomagnetic-shielding effect on energetic particles is often quantified via the VADM value of M. The modern (epoch 2000) dipole moment is M0 = 7.8·1022 A·m2. Here we normalize the apparent 14C production Q during different events to the standard modern conditions, viz. the production that would be if the event took place nowadays. For that, we modeled the dependence of Q on M using a broad range of SEP spectra measured during the recent decades59. The spectra are bound between the hardest known spectrum of SEP event on 23-Feb-1956 (GLE 0560) and the softest strong-event spectrum of 04-Aug-1972 (GLE 2460), as shown in Supplementary Fig. 4a. The same plot shows also the global yield function, denoted as Yp(E), of 14C for three VADM values of M = 3, 6 and 9 (x1022 A·m2). The global production function is defined as

$${Y}_{p}(E,M)=\int _{0}^{\pi /2}Y(E)\cdot H(E,M,\theta )\cdot \,\sin (\theta )\cdot d\theta$$
(10)

where Y(E) is the columnar yield function of 14C by protons with energy E in the Earth’s atmosphere61, H(E) is a step-like magnetospheric transmissivity function taking the value of 0 for E < Ec and 1 otherwise, where Ec is the energy corresponding to the local geomagnetic rigidity cutoff defined by the dipole moment M and the polar angle (co-latitude) θ. The production is higher for a weaker geomagnetic field (smaller values of M) and vice versa. The production of 14C by a specific SEP event with a given proton spectrum Jp is defined as the integral of the product of the yield function and the energy spectrum of particles.

$$Q(M)=\int _{0}^{\infty }{Y}_{p}(E,M)\cdot {J}_{p}(E)\cdot dE$$
(11)

The dependence of Q on M is shown in Supplementary Fig. 4b for the two analyzed SEP spectra. In order to compensate for the variable geomagnetic field, we introduce the correction factor, which relates the 14C production at the VADM M to that at the reference field M0:

$$C=\frac{Q({M}_{0})}{Q(M)}$$
(12)

The correction factor is shown in Supplementary Fig. 4c for the two SEP spectra. Although the spectra are essentially different (panel a) and the isotope production for the two events is different too (panel b), the correction factor appears robust against the exact spectral shape. This correction factor was further applied to compare the strength of all studied events to the standard conditions, as shown in Table 1.

Reconstructions of the geomagnetic field in the past is a difficult task, and the result can be uncertain. To cover a possible range of VADM values we took a conservative approach and considered two paleomagnetic reconstructions for the last millennia, as shown in Supplementary Fig. 5: a recent GGF100k paleomagnetic model by Panovska et al.40 and a reconstruction by Knudsen et al.41 based on the GEOMAGIA50 database. Although there are many other paleomagnetic reconstructions, these two cover the full range of uncertainties.

The uncertainties of the corrected for the VADM changes 14C production (last two columns in Table 1) are mostly defined by the uncertainties of the actual production (column 3) and VADM uncertainties (columns 4 and 5). The resultant uncertainty of the corrected production was assessed using the Monte–Carlo method by randomly picking pairs of M and Q from the normally distributed values within the ranges shown in Table 1 and computing the mean and the standard deviation of the corrected Q values.


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# Problem of the Week Problem A and Solution School Supply Spending

## Problem

Mr. Cameron has $100 to spend on school supplies for his classroom. His students need pencils, erasers, binders, lined paper, whiteboard markers, and crayons. When he looks through the catalogue, he discovers that the items he needs are only available in packages. The table below shows the prices. Supplies Amount per Package Cost per Package pencils 10$0.50
erasers 12 $1.00 binders 25$5.00
lined paper 100 $1.50 whiteboard markers 4$4.00
crayons 24 $1.00 Mr. Cameron has 20 students in his class. Each student needs 5 pencils, 2 erasers, 5 binders, 50 sheets of lined paper, 1 whiteboard marker, and 1 entire package of crayons. Mr. Cameron needs 8 whiteboard markers himself. 1. Does Mr. Cameron have enough money to buy all of his classroom supplies? Justify your answer. 2. If he does have enough money, how much is leftover from the$100? If he does not have enough money, how much more would he need?

## Solution

A) To solve this problem, we need to determine how many packages of each supply are needed in order to have enough for all 20 students in Mr. Cameron’s class.

Since there are 10 pencils per package and each student requires 5 pencils, then 1 package of pencils is enough for $$10 \div 5 = 2$$ students. Since there are 20 students in the class, Mr. Cameron needs $$20 \div 2 = 10$$ packages of pencils.

Alternatively, since there are 20 students, and each student needs 5 pencils, then Mr. Cameron needs $$20 \times 5 = 100$$ pencils. Since each package has 10 pencils, he needs $$100 \div 10 = 10$$ packages of pencils.

Similarly, we can calculate:

• 1 package of binders is enough for $$25 \div 5 = 5$$ students, so he requires $$20 \div 5 = 4$$ packages of binders.
OR
Mr. Cameron needs a total of $$20 \times 5 = 100$$ binders, which means he requires $$100 \div 25 = 4$$ packages.

• 1 package of lined paper is enough for $$100 \div 50 = 2$$ students, so he requires $$20 \div 2 = 10$$ packages of lined paper.
OR
Mr. Cameron needs a total of $$20 \times 50 = 1000$$ sheets of paper, which means he requires $$1000 \div 100 = 10$$ packages.

Also, 1 package of erasers is enough for $$12 \div 2 = 6$$ students. This means we need $$20 \div 6 = 3$$ packages with a remainder of 2. So 3 packages would not have enough erasers for the class, and thus we need to round up to 4 packages.

Alternatively, we need $$20 \times 2 = 40$$ erasers, which means we require $$40 \div 12$$ packages, which is equal to 3 packages with a remainder of 4, and thus we round up to 4 packages. Both calculations determine that we require 4 packages of erasers. Note that the remainders are different, because in the first division we are calculating a number of students and in the second division we are calculating a number of erasers.

Since each student needs just 1 whiteboard marker, we require $$20 \div 4 = 5$$ packages for the students. However, since Mr. Cameron needs 8 markers himself, that would require 2 more for a total of 7 packages of markers.

Alternatively, we need $$20 + 8 = 28$$ whiteboard markers in total, which means we require $$28 \div 4 = 7$$ packages.

Finally, we need 1 package of crayons for each student, which is a total of 20 packages.

Now we can make a new table knowing how many packages of each supply are required for the class. We can multiply the number of packages by the cost per package for each of the supplies.

Supplies Number of Packages Cost per Package (in $) Total Cost (in$)
pencils 10 0.50 $$10 \times 0.50 = 5.00$$
erasers 4 1.00 $$4 \times 1.00 = 4.00$$
binders 4 5.00 $$4 \times 5.00 = 20.00$$
lined paper 10 1.50 $$10 \times 1.50 = 15.00$$
whiteboard markers 7 4.00 $$7 \times 4.00 = 28.00$$
crayons 20 1.00 $$20 \times 1.00 = 20.00$$

So the total cost of supplies is: $5.00 + 4.00 + 20.00 + 15.00 + 28.00 + 20.00 = \92.00$

B) Since Mr. Cameron has \$100 to spend, he has enough for these supplies with $$\100 - \92 = \8$$ left over.

### Teacher’s Notes

When we are solving this problem, essentially we need to find equivalent ratios. We have been told the ratio of one student to a number of pencils, erasers, binders, sheets of paper, whiteboard markers, and crayons. We need an equivalent ratio for 20 students to amounts of each of those items.

For example, with pencils we want a ratio of $$1:5$$, so we need an equivalent ratio $$20:x$$.
Now we just need to solve for $$x$$.

One technique for finding equivalent ratios is to rewrite the ratios as fractions.
For example, the ratio $$1:5$$ can be written as the fraction $$\tfrac{1}{5}$$.
So now we just need to solve the equation:

$\dfrac{1}{5} = \dfrac{20}{x}$

If we have two fractions that are equal, that means the products of the numerator of one and the denominator of the other are equal. This is sometimes referred to as cross-multiplying. In this example it means:

$1 \times x = 5 \times 20$

So,

$x = 100$

This means we need 100 pencils to supply the whole class.


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### Home > CALC > Chapter 10 > Lesson 10.1.8 > Problem10-80

10-80.

Use substitution. Let u = tan(t).

Use partial fraction decomposition.

1/(x – 2) is undefined when x = 2, so a limit needs to be used to evaluate this integral.

Since this is the integral of a derivative, your answer should have a "+ C".

Use substitution. Let u = ln(x).