Question

A man $6^{\prime }$ tall moves away from a source of light ${20}^{\prime }$ above the ground level, his rate of walking being $4m.p.h$. At what rate is the tip of his shadow moving?

• A. $\frac{12}{7}$
• B. $\frac{3}{7}$
• C. $\frac{40}{7}$
• D. None of these

Answer 1

The correct option is C $\frac{40}{7}$

Let $x$ be the distance between the man and the street light and let $s$ be the length of his shadow.

GIVEN: $\frac{dx}{dt}=4$.

Distance of tip: $s+x$

Therefore rate of tip: $\frac{ds}{dt}+\frac{dx}{dt}$

Let us find $\frac{ds}{dt}$ when $x=30$.

Note that we have two similar triangles: the overall triangle with height $20$ and base length $s+x$ and the smaller one inside of it with height $6$ and base length $s$. We want to solve for $s$ in terms of $x$ by using the ratio

$\frac{6}{s}=\frac{20}{s+x}$

$\Rightarrow 6(s+x)=20s$

$\Rightarrow 6s+6x=20s$

$\Rightarrow 14s=6x$

$\Rightarrow s=\frac{6}{14}x$

$\Rightarrow s=\frac{3}{7}x$

Differentiate both sides we get

$\frac{ds}{dt}=\frac{3}{7}\frac{dx}{dt}$

$\frac{ds}{dt}=\frac{12}{7}(\because \frac{dx}{dt}=4)$.

Therefore rate of tip: $\frac{ds}{dt}+\frac{dx}{dt}=\frac{12}{7}+4=\frac{40}{7}$