Question

A man of height 6 ft walks at a uniform speed of 9 ft/sec from a lamp fixed at 15 ft height. The length of his shadow is increasing at the rate of
(a) 15 ft/sec
(b) 9 ft/sec
(c) 6 ft/sec
(d) none of these

Answer 1

(c) 6 ft/sec

Let AB be the lamp post. Suppose at any time t, the man CD be at a distance of x km from the lamp post and y ft be the length of his shadow CE.


$$\begin{gathered} \mathrm{Since}\mathrm{the}\mathrm{triangles}ABE\mathrm{and}CDE\mathrm{are}\mathrm{similar}, \\ \frac{AB}{CD}=\frac{AE}{CE} \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{15}{6}=\frac{x+y}{y} \\ \Rightarrow \frac{x}{y}=\frac{15}{6}-1 \\ \Rightarrow \frac{x}{y}=\frac{3}{2} \\ \Rightarrow y=\frac{2}{3}x \\ \Rightarrow \frac{dy}{dt}=\frac{2}{3}\left(\frac{dx}{dt}\right) \\ \Rightarrow \frac{dy}{dt}=\frac{2}{3}\times 9 \\ \Rightarrow \frac{dy}{dt}=6\mathrm{ft}/\sec \end{gathered}$$