Question

A man $1.6m$ tall walks at the rate of $0.5m/s$ away from a lamp post of $8$ meters height. find the rate at which his shadow is increasing.

Answer 1

Consider the fig,

Let $x$ be the distance of the man from the pole at time $t$.

Let $s$ be the length of the shadow at time $t$.

$△CED\sim △AEB$ from $AAA$ Postulate

$⟹\frac{CE}{CD}=\frac{AE}{AB}$

$\frac{s}{1.6}=\frac{x+s}{8}$

$5s=x+s$

$s=\frac{1}{4}x$

The man is walking at the speed of $0.5m/s$

$\frac{dx}{dt}=0.5$

Rate of change of length of the shadow is $\frac{ds}{dt}$

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

$=\frac{d}{dx}\left(\frac{1}{4}x\right)0.5$

$=\frac{1}{4}\times 0.5=0.25$

$\therefore$ Rate at which the shadow is increasing is $0.25m/s$