Solving exponential simultaneous equations

[MathsDude69]

Homework Statement

The actual problem shows a graph however I can state all the information. The graph is of a sinusiodal waveform where the amplitude is decaying exponentially. The formula for the graph is given by the equation:

T = Ae^-Ktsin(wt + ø)

The question is to find A,K,w and ø

Being quite confident in sinusoidal waveforms I can tell you that:

w = 40 x pi or 125.66 (whichever tickles your fancy)
ø = -1.885

However I am stuck with the A and K.

Assuming that the maximum peaks occur when sin(wt + ø) = 1 then:

0.23 = Ae^-K0.0275

0.08 = Ae^-K0.0775

I now have 2 points to solve simultaneously for A and K.

The Attempt at a Solution

0.23/0.08 = Ae^-K0.0275/Ae^-K0.0775

2.875 = e^-K0.5

K = (1/0.5) x ln(2.875) = 2.1121


When you plug this back into the two equations however you get two different answers for A and A is supposed to be a constant. Can anyone see where I am goign wrong here?

Thanks in advance for any help.

 

[yeongil]

0.23/0.08 = Ae^-K0.0275/Ae^-K0.0775

2.875 = e^-K0.5
K = (1/0.5) x ln(2.875) = 2.1121

Oops! That should be e^0.05k in the second line. When dividing two powers you subtract the exponents. Also, if we pretend that the 2nd line was right, then the 3rd line is missing a negative in front of the fraction.

But anyway, there should be no negative:

$$2.875 = e^{0.05k}$$

$$k = \frac{ln(2.875)}{0.05} = 21.121$$

Now you should get a single value for A.


01

 

[Architeuthis Dux]

Hi there,

It seems like you are on the right track for solving these exponential simultaneous equations. However, I believe the issue lies in your assumption that the maximum peaks occur when sin(wt + ø) = 1. While this may be true in some cases, it is not always the case for exponential decay.

Instead, I would suggest using the fact that T is a decaying exponential function, meaning that as t increases, T decreases. This means that the maximum value of T occurs at t = 0, and we can use this to solve for A and K.

First, let's set t = 0 in the equation T = Ae-Ktsin(wt + ø). This gives us:

T(0) = Ae^0sin(w(0) + ø)

T(0) = A(0)sin(ø)

T(0) = 0

This means that the maximum value of T is 0, and we can use this to solve for A and K in the two equations you wrote:

0.23 = Ae-K0.0275

0.08 = Ae-K0.0775

Now, let's substitute in 0 for T(0) in these equations:

0.23 = A(0)e-K0.0275

0.08 = A(0)e-K0.0775

We can see that the only unknown in these equations is A(0), so we can solve for it by dividing the second equation by the first:

0.08/0.23 = e-K(0.0775-0.0275)

0.3478 = e-K0.05

K = (1/0.05)ln(0.3478) = -2.0432

Now, we can plug this value for K into either of the original equations to solve for A:

0.23 = Ae-(-2.0432)0.0275

0.23 = A(0.9769)

A = 0.2357

Therefore, our final solution is:

A = 0.2357
K = -2.0432
w = 40 x pi or 125.66 (whichever you prefer)
ø = -1.885

I hope this helps and clarifies any confusion you had. Good luck with your problem!

 

[Architeuthis Dux]

I would first commend the student for their understanding and attempt at solving the problem. Solving exponential simultaneous equations can be challenging, but with practice and careful analysis, it can be done accurately.

To address the issue of two different answers for A, it is important to remember that when solving simultaneous equations, we are looking for the values that satisfy both equations simultaneously. In this case, the two equations represent two points on the same sinusoidal waveform, so the values for A and K should be the same for both equations.

One possible explanation for the discrepancy could be a calculation error. It is important to double check all calculations and make sure they are accurate. Another possibility is that the student may have made a mistake in the assumption that the maximum peaks occur when sin(wt + ø) = 1. This may not always be the case, especially if the waveform is decaying exponentially.

In order to accurately solve for A and K, it may be helpful to use a different approach. One method could be to rewrite the equations in terms of logarithms, which can make solving for exponential variables easier. Another approach could be to use a graphing calculator to plot the two equations and find the intersection point, which would give the values for A and K.

Overall, solving exponential simultaneous equations requires careful analysis, attention to detail, and possibly trying different methods to find the most accurate solution. With practice and perseverance, the student will be able to successfully solve these types of problems.