Question

A man of height $h$ is walking away from a street lamp with a constant speed $v$. The height of the street lamp is $3h$. The rate at which length of the man's shadow is increasing when he is at a distance $10h$ from the base of the street lamp is

• A. $\frac{v}{2}$
• B. $\frac{v}{3}$
• C. $2v$
• D. $\frac{v}{6}$

Answer 1

The correct option is A $\frac{v}{2}$

Let the horizontal distance between lamp and person be $a$ and the horizontal distance between person and shadow be $b$
From the figure:
$\Delta ABC$ and $\delta ADE$ are similar

$\frac{DE}{BC}=\frac{AE}{AC}$

$\Rightarrow \frac{3h}{h}=\frac{a+b}{b}$

$\Rightarrow 3b=a+b$

$\Rightarrow a=2b$

Differentiating on both sides w.r.t. $t$
$\frac{\partial a}{\partial t}=\frac{\partial 2b}{\partial t}$

$\Rightarrow v_{man}=2\times v_{shadow}$

$\Rightarrow v_{shadow}=\frac{v}{2}$