Solving exponential simultaneous equations

In summary, the conversation is about finding the values of A, K, w, and ø for a sinusoidal waveform given by the equation T = Ae-Ktsin(wt + ø). The question is to find these values and the person is confident in finding w and ø but is stuck with A and K. They use two points to solve simultaneously for A and K, but when they plug in a value for K, they get two different answers for A. After realizing a mistake in their calculation, they correct it and find that the value for K is actually 21.121. This leads them to a single value for A.
  • #1

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.
 
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  • #2
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 e0.05k 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:

[tex]2.875 = e^{0.05k}[/tex]

[tex]k = \frac{ln(2.875)}{0.05} = 21.121[/tex]

Now you should get a single value for A.


01
 
  • #3



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!
 
  • #4


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.
 

1. How do I solve exponential simultaneous equations?

To solve exponential simultaneous equations, you can use the substitution method or the elimination method. Both methods involve manipulating the equations to eliminate one of the variables and then solving for the remaining variable.

2. What is the difference between exponential and linear simultaneous equations?

The main difference between exponential and linear simultaneous equations is that exponential equations involve variables raised to a power, while linear equations do not. This makes the process of solving exponential simultaneous equations more complex compared to solving linear simultaneous equations.

3. Can I use a graphing calculator to solve exponential simultaneous equations?

Yes, you can use a graphing calculator to solve exponential simultaneous equations. Most graphing calculators have a feature that allows you to input and solve systems of equations, including exponential equations.

4. Are there any special cases when solving exponential simultaneous equations?

Yes, there are some special cases that may arise when solving exponential simultaneous equations. These include equations with no solution, infinitely many solutions, or equations that can be simplified to a linear equation.

5. How can I check my solutions to exponential simultaneous equations?

You can check your solutions by substituting them into both equations and verifying that they satisfy both equations. You can also graph the equations and see if the intersection point matches your solutions.

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