At a concert, a band is playing on a platform that extends P P feet from a wall behind the band, and the platform is rising from ground level.

Question

A light source is L feet from the wall, and the platform casts a lengthening shadow on the wall as the platform rises. At time t seconds, the platform is h feet above the ground, and the height of the shadow is s feet. The quantities are related by the equation (1/L)(H+S)=(1/P)S, where L and P are constants. Which of the following best expressed the rate of change of h with respect to time in terms of the rate of change of s with respect to time.

a. dh/dt = (L/P)s-s

b. dh/dt = (L/P)s - (ds/dt)

c. dh/dt = (L/P)(ds/dt) - s

d. dh/dt = (L/P) (ds/ds) - (ds/dt)

Answer 1

Given: (1/L) (h + s) = (1/P) s . . . . . . this is simply from the fact that the big triangle from light to wall is

similar to the small triangle which includes the lengths s and P ;-)

h + s = (L/P) s . . . . . . . multiplying through by L

h = (L/P) s - s

=> dh/dt = (L/P ds/dt - ds/dt . . . differentiating through w.r.t. t

As presented above, none of the answers is correct. :-(

If there is a small typo in answer (d) and it should read: dh/dt = (L/P) (ds/dt) - (ds/dt), then (d) is correct.