Integrals and Related Rates

[superbasel97]

#2 (2 pts) A window in Prof. Xu' ss office is shown in figure (A) below,  
y = f ( s )  
where f is given by f (s) = 5 sin ((Ï€/32)(s^2 + 16)).  

Every evening, before she leaves her office, Prof. Xu draws  
the curtain (see figure (B)) over the window.  
Every morning she draws the curtain open, starting at time t = 0.  
If she draws the curtain at the rate of 0.8 ft / sec,  
find the rate at which the area of the exposed part of the window' s glass  
is changing when t = 4.  

 

 

​not sure where to start, dont need answer needing guiding help thx

 

[kw0573]

A diagram would be helpful.
If s is the horizontal distance, then 

1) Using speed = distance / time. Write an expression for time t in terms of s. Determine the s covered from t = 0 to t = 4. (again a diagram here helps to know if Prof Xu opens the window from one side or from the middle.)
2) Write an expression for the area of the exposed window for the bounds of s determined from part 1).
3) Apply a method to rewrite the area expression, now in terms of t instead of s. Remember to change your bounds.
4) What operation would you do to find the rate of change of the expression in 3)? How can you simplify your calculations? You don't actually need to do any calculus if you have to do both 3 and 4. Hint: a conclusion from the fundamental theorem of calculus says that if we take the derivative of an integral function that goes from 0 to a, is the original function evaluated at x = a (or in this question, t = a). 
https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Examples

Hope that helps!

Edited December 4, 2015 by kw0573