Question

A man of height $1.8$meters is moving from a lamp post at the rate of $1.2m/sec$. If the height of the lamp post is $4.5$ metres, then the rate at which the shadow of the man is lengthening is

• A. $0.4m/sec$
• B. $0.8m/sec$
• C. $1.2m/sec$
• D. None of these

Answer 1

The correct option is B

$0.8m/sec$

Explanation for the correct option:

Step 1. Finding the rate at which the shadow of the man is lengthening:

Let PQ be the man and AB be the Lamp post. Let CQ be the shadow.

Since, the man is moving from a lamp post at the rate of $1.2m/sec$. Therefore,

$\frac{dy}{dt}=1.2$

According to the figure,

$\begin{array}{rcl}\frac{4.5}{1.8}& =& \frac{x+y}{x}\end{array}$

$\Rightarrow$$\begin{array}{rcl}\frac{5}{2}& =& \frac{x+y}{x}\end{array}$

$\Rightarrow$$\begin{array}{rcl}5x& =& 2x+2y\end{array}$

$\Rightarrow$$\begin{array}{rcl}3x& =& 2y\end{array}$

$\Rightarrow$ $\begin{array}{rcl}x& =& \frac{2}{3}y\end{array}$

Step 2. Differentiate it with respect to $\begin{array}{l}t\end{array}$

$\begin{array}{rcl}\frac{dx}{dt}& =& \frac{2}{3}\left(\frac{dy}{dt}\right)\\ & =& \frac{2}{3}\times 1.2\\ & =& 0.8\end{array}$

$\therefore$Required rate of length of shadow is $\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

Hence, the correct answer is option B