Question

A street light is $6m$ high. A man $180cm$ tall goes away from the street light at the speed of $120cm/sec$. Find the rate of increase of length of his shadow when he is $9m$ away from the light. Also find the rate at which the tip of the shadow moves on the road.

Answer 1

Street light Height $=6m=600cm$

Man Height $=180cm$

Man speed $=120cm/sec$

Let the Distance of men from height $=x$

Distance of length of shadow from men $=y$

as $△ABC\sim △PQC$

$\therefore \frac{AB}{BC}=\frac{PQ}{QC}$

$\frac{600}{x+y}=\frac{180}{y}$

$600y=180x+180y$

$180x=420y,y=\frac{3}{7}x$

$\frac{d{y}^{\prime }}{dt}=\frac{3}{7}\frac{dx}{dt}$

$(\frac{dx}{dt}=120)$

$=\frac{3}{7}\propto 120=\frac{360}{7}=51.4cm/sec$

Rate of increase of shadow $=57.4cm/sec$