## 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 5ft/s along a straight path. How fast is the tip of the shadow moving when he is 40 ft from the pole?

Sep 25, 2016

The tip of the shadow is moving at a speed of $\frac{25}{3} = 8. \overline{3} \text{ft"/"s}$

#### Explanation:

First, let's sketch the situation: In the above image, $m$ is the distance from the pole to the man, and $s$ is the distance from the pole to the tip of the man's shadow. Our goal is to find the rate of change of $s$ with respect to time given that rate of change of $m$ with respect to time is $5 \text{ft"/"s}$ and $m = 40 \text{ft}$

As derivatives are rates of change, we can rewrite our goal as trying to find $\frac{\mathrm{ds}}{\mathrm{dt}}$ given $\frac{\mathrm{dm}}{\mathrm{dt}} = 5$ and $m = 40$.

By the properties of similar triangles, we have

$\frac{s}{15} = \frac{s - m}{6}$

$\implies 2 s = 5 s - 5 m$

$\implies s = \frac{5}{3} m$

Differentiating with respect to time, we get

$\frac{\mathrm{ds}}{\mathrm{dt}} = \frac{5}{3} \frac{\mathrm{dm}}{\mathrm{dt}} = \frac{5}{3} \cdot 5 = \frac{25}{3}$

As it so happens, the rate of change of the tip of the shadow with respect to time is independent of the value of $m$, and our final result is that the tip of the shadow is moving at a speed of $\frac{25}{3} = 8. \overline{3} \text{ft"/"s}$