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.


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