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 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?

Answer 1

`20/3 (ft)/s`

Explanation:

in this diagram, x is the distance from the man to the pole, and y is the distance from the tip of the man's shadow to the pole. i assume the man and pole are standing straight up, which means the 2 triangles are similar.
by similarity, `(y-x)/y=6/15`
`15(y-x)=6y`
`15y-15x=6y`
`9y=15x`
`y=5/3x`

differentiate both sides with respect to `t` or time.
`dy/dt=5/3dx/dt`

you know `dx/dt=4(ft)/s` because the man is walking that speed away from the pole. you want to find `dy/dt`, how fast the tip of the shadow is moving.

that means `dy/dt=5/3*4(ft)/s=20/3(ft)/s`

actually, the man's distance from the pole doesn't matter since only his speed affects how fast his shadow moves.