Poles street lights

Question

A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 45 ft from the pole?

Answer 1

This problem requires some calculus to solve.

First make a sketch.

I drew a pole with a man in front of it and labeled their heights.

Then I drew a line connecting the top of the pole - to the top of the man - to the ground (the shadow).

I marked the distance between the pole and the man as x

and the distance from the pole to the shadow as y.

This involves similar triangles since the triangle formed between the pole and the ground and the man and the ground share the same angle.

This mean the ratios of the sides are equal.

15/y = 6/(y-x). y-x is the distance between the shadow's edge and the man.

Simplifying you get y = 5/3 x

Now you want to find the speed the shadow relative to the pole. This is dy/dt

d/dt (y) = d/dt (5/3 x)

dy/dt = 5/3 dx/dt

We know dx/dt is the speed of the man or 5ft/s

This can be used to find the speed of the shadow.

Oddly in this problem the distance the man is from the pole is completely irrelevant.