Question

A man of $2$ m height walks at a uniform speed of $6$ km/h, away from a lamp post of $6$ m height. The rate at which the length of his shadow increase is

• A. $2$ km/h
• B. $1$ km/h
• C. $3$ km/h
• D. $6$ km/h

Answer 1

The correct option is C

$3$ km/h

Finding the rate at which the length of the shadow will increase:

Let the length of the shadow be ‘$y$’ m and the distance from the lamp post and the man be ‘$x$’ m.

$$\begin{gathered} AB=6cm \\ CD=2cm \end{gathered}$$

By the theorem of similar triangles,

$\begin{array}{rcl}\frac{AB}{BE}& =& \frac{CD}{DE}\\ \frac{6}{x+y}& =& \frac{2}{y}\\ 6y& =& 2x+2y\\ x& =& 2y\\ \frac{dx}{dt}& =& 2\frac{dy}{dt}\\ 6& =& 2\frac{dy}{dt}\left(\because \frac{dx}{dt}=6\right)\\ \frac{dy}{dt}& =& 3km/h\end{array}$

The rate at which the length of his shadow increase is $\mathbf{3}$ km/h.

Hence, the correct option is (C).