cal question?

Question

A street light is at the top of a 16.5 ft. tall pole. A man 5.6 ft tall walks away from the pole with a speed of 4.5 feet/sec along a straight path. How fast is the tip of his shadow moving away from the pole when the man is 31 feet from the pole?

Answer 1

y/16.5 = (y - x)/5.6
5.6y = 16.5y - 16.5x
10.9y = 16.5x
dy/dt = (16.5/10.9)dx/dt
dy/dt = 16.5*4.5/10.9 = 6.81 ft/s

Answer 2

shadow length......man's height
--------------------- = --------------------
pole to tip...............pole's height

y - x.......5.6
------- = ------ where x is pole to man
y..........16.5

16.5y - 16. 5x = 5.6y
10.9y = 16.5x
y = (16.5 / 10.9)x

y(t) = (16.5 / 10.9)x is the distance from pole to tip of the shadow when the man is x feet from the pole

Differentiate y(t) with respect to time t to find the rate at which the shadow's length is increasing
dy / dt = (16.5 / 10.9) dx / dt

Remember that the man's rate is dx / dt = 4.5 ft / s.

dy / dt = (16.5 / 10.9)(4.5 ft / s) = 6.8 ft / s

Answer: 6.8 ft / s