Question in Calculus

Question

A street light is at the top of a 12 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 50 feet from the base of the pole?

The tip of the shadow is moving at 

Answer 1

Let x is distance from bottom of light. Then if length of shadow is l we have (x + l)/l = 12/6 = 2; x/l = 1; l = x

dl/dt = dx/dt = 5 ft/s. So, tip of shadow has constant speed 5 ft/s

Answer 2

If D = the distance from the pole and L = the length of the shadow, it can be shown that

L = D, which means that

dL/dt = dD/dt

Since dD/dt = 5 ft/sec at any distance from the pole

dL/dt = 5 ft/sec at any distance from the pole

From the pole, her shadow is moving 5 + 5 = 10 ft/sec.

Where L = D comes from. The larger triangle and the smaller triangle contained within are similar,

so (L +D)/L = 1+ D/L = 12/6 = 2.

This means D/L = 1. So L = D.