Question

A street light is at the top of a pole that is 16 feet tall. A woman 6 ft tall walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the length of her shadow moving when she is 45 ft from the base of the pole?

Answer 1

If you say x = distance from pole to the woman and z = distance from the pole to the tip of her shadow, from similar triangles, you can write:
z/16 = (z-x)/6. (1)
We also know that the woman is moving at 7 feet/second or dx/dt = 7
You need to find dz/dt when x = 45. When you differentiate equation (1) with respect to t, you get:
(1/16)*dz/dt = (1/6)(dz/dt - dx/dt), or
6*dz/dt = 16(dz/dt - dx/dt), or 10*dz/dt = 16*dx/dt or dz/dt = 1.6*dx/dt
which means the distance 45 does not enter into the equation.
Plug dx/dt = 7, you get dz/dt = 1.6*7 = 11.2 feet/sec.