A light is on the top of a 12 ft tall pole and a 5ft 6in tall person is walking away from the pole at a rate of 2 ft/sec.

Question

1. At what rate is the tip of the shadow moving away from the pole when the person is 25 ft from the pole?
2. At what rate is the tip of the shadow moving away from the person when the person is 25 ft from the pole?

Answer 1

Draw a picture:

Using similar triangles, we can say:

Cross multiplying, we get:

12(l - x) = 5.5l

12l - 12x = 5.5l

6.5l = 12x

x = (6.5/12)l

Taking the derivative with respect to time we get:

dx/dt = (6.5/12)dl/dt or

dl/dt = (12/6.5)(dx/dt)

We know dx/dt = 2 so dl/dt = (12/6.5)(2) = 24/6.5 = 48/13 ft/sec

For question 2, just subtract the rate the shadow is going from the rate the man is going so 48/13 - 2 = 22/13 ft/sec.

Notice that the 25 foot distance doesn't matter in this problem.