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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
Options
1.6 km/hr
6.3 km/hr
5 km/hr
3.2 km/hr
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Solution
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.
\[\text { Since triangles ABE and CDE are similar }, \]
\[\frac{AB}{CD} = \frac{AE}{CE}\]
\[\Rightarrow \frac{5}{2} = \frac{x + y}{y}\]
\[ \Rightarrow \frac{x}{y} = \frac{5}{2} - 1\]
\[ \Rightarrow \frac{x}{y} = \frac{3}{2}\]
\[ \Rightarrow y = \frac{2}{3}x\]
\[ \Rightarrow \frac{dy}{dt} = \frac{2}{3}\left( \frac{dx}{dt} \right)\]
\[ \Rightarrow \frac{dy}{dt} = \frac{2}{3} \times 4 . 8\]
\[ \Rightarrow \frac{dy}{dt} = 3 . 2 \ km/hr\]
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