Question

A man 2 metres tall walks away from a lamp post 5 metres height at the rate of 4.8 km/hr. The rate of increase of the length of his shadow is
(a) 1.6 km/hr
(b) 6.3 km/hr
(c) 5 km/hr
(d) 3.2 km/hr

Answer 1

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 m be the length of his shadow CE.



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

$$\begin{gathered} \Rightarrow \frac{5}{2}=\frac{x+y}{y} \\ \Rightarrow \frac{x}{y}=\frac{5}{2}-1 \\ \Rightarrow \frac{x}{y}=\frac{3}{2} \\ \Rightarrow y=\frac{2}{3}x \\ \Rightarrow \frac{dy}{dt}=\frac{2}{3}\left(\frac{dx}{dt}\right) \\ \Rightarrow \frac{dy}{dt}=\frac{2}{3}\times 4.8 \\ \Rightarrow \frac{dy}{dt}=3.2\mathrm{km}/\mathrm{hr} \end{gathered}$$