Find the maximum amplitude of a dampened swinging door

[Bluewolf1986]

Here's a question from my Calculus 1: Differential Calculus Course that I'm struggling with:

The angle A of a heavily damped swinging door varies with time according to the law:


A=e^(-at)-b^(-bt)/(b-a)

at what time (t) is the amplitude the greatest?


(Enter your answer in terms of e, and the parameters a and b. Type ∗ for multiplication; / for division; ∧ for exponentiation. The functions sqrt(x), ln(x), sin(x), etc. are known. Type e and pi for the mathematical constants e and π.)

Here is my work so far to try to find the maximum amplitude like the maximum height which is not right as I'm confused with this angular problem:
1) A'(t)= A'(0)= -ae^(-at)-(-be^(-bt)/(b-a)
= -a+b/b-a

2) -a+b/b-a=1 second

Please help me with suggested a method to find the answer in terms of e and π

Thanks.

 

[Deleted member 4993]

What is the derivative of:

y = \(\displaystyle \frac{b^{-b*t}}{b-a}\)

 

[tkhunny]

1) How does A(t), the ANGLE of the door, relate to Amplitude? They are not the same thing.
2) Why did you pick t = 0? You're assuming the maximum is at the beginning? And what maximum is that, Angle or Amplitude?

 

[Bluewolf1986]

1) How does A(t), the ANGLE of the door, relate to Amplitude? They are not the same thing.
2) Why did you pick t = 0? You're assuming the maximum is at the beginning? And what maximum is that, Angle or Amplitude?

1) I'm not sure, but I think amplitude is related to angle in that it is equal to 1/2 of a vibration path, which would use some function to calculate. I'm not sure which one though, my guess would be sin(x) or cos(x) given the vibrational wave connection. Although I'm confused about Euler's number being involved in the calculation as I was never taught to solve problems using both trig and exponential functions.

2) I was hoping to find the maximum after the beginning, although I'm not sure how to solve for the maximum point except for projectile motion/ freefall problems. So using those principles I first attempted to solve for the maximum Amplitude.

 

[Deleted member 4993]

In your calculus class, (I am sure) there must have been discussion about derivative of a function (related to its dependent variable) and the relation (value) of the derivative at the local extrema.

What was the relation?

 

[tkhunny]

1) I'm not sure, but I think amplitude is related to angle in that it is equal to 1/2 of a vibration path, which would use some function to calculate. I'm not sure which one though, my guess would be sin(x) or cos(x) given the vibrational wave connection. Although I'm confused about Euler's number being involved in the calculation as I was never taught to solve problems using both trig and exponential functions.

Time to think it through. What is the angle doing when the door is at its maximum deflection? Is it when the angle is zero (0), shut, or at some other time.

2) I was hoping to find the maximum after the beginning, although I'm not sure how to solve for the maximum point except for projectile motion/ freefall problems. So using those principles I first attempted to solve for the maximum Amplitude.

What does projectile motion have to do with the workings of the derivative? It's a function. It has changes. The 1st derivative can help you find where it changes direction. The derivative doesn't work differently just because it's a different sort of problem.

 

[brmjerry]

What is the derivative of:

y = \(\displaystyle \frac{b^{-b*t}}{b-a}\)

the derivative of the function is the angular velocity in radians
[MATH]\omega=\dfrac{d}{dt}A= \dfrac{1}{b-a}\dfrac{d}{dt}(e^{-at}-e^{-bt})=\dfrac{1}{b-a}(-ae^{-at}+be^{-bt})[/MATH]which has a root at ln(b/a) which seems to be the maximum amplitude
but they want the answer in answer in terms of e, a and b
the door is closed at t = 0
thank you for your time