Question

Help?
A man 2 meters tall walks at the rate of 2 meters per second toward a streetlight that's 5 meters above the ground. At what rate is the tip of his shadow moving?

Answer 1

Let \(s\) be the length of the shadow, so that \(y=x+s\) in the picture. From triangle similarity, you have the ratio
\[\frac{s}{2}=\frac{x+s}{5}~~\implies~~\frac{3}{2}s=x\]
Differentiating both sides with respect to \(t\) gives
\[\frac{3}{2}\frac{ds}{dt}=\frac{dx}{dt}\]
Given that \(\dfrac{dx}{dt}=-2\dfrac{\text{m}}{\text{s}}\), solve for \(\dfrac{ds}{dt}\).

Answer 2

How would I do that?

Answer 3

Plug it in:
\[\frac{3}{2}\color{red}{\frac{ds}{dt}}=-2~~\implies~~\frac{ds}{dt}=\cdots\]

Answer 4

Gottcha!