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 increases is
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 increases is
Updated On: Jun 7, 2024
- $2\, km/h$
- $1\, km/h$
- $3\, km/h$
- $6\, km/h$
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The Correct Option is C
Solution and Explanation
In $ \Delta ADC, $ $ \tan \theta =\frac{6}{x+y} $ and in $ \Delta BCE, $
$ \therefore $ $ \frac{2}{x}=\frac{6}{x+y}\Rightarrow x+y=3x $
$ \Rightarrow $ $ y=2x $
On differentiating w.r.t. t, we get
$ \frac{dy}{dt}=2\frac{dx}{dt} $
$ \Rightarrow $ $ 6=2\frac{dx}{dt} $ $ \left( \because \frac{dy}{dt}=6given \right) $
$ \Rightarrow $ $ \frac{dx}{dt}=3\,km/h $
$ \therefore $ $ \frac{2}{x}=\frac{6}{x+y}\Rightarrow x+y=3x $
$ \Rightarrow $ $ y=2x $
On differentiating w.r.t. t, we get
$ \frac{dy}{dt}=2\frac{dx}{dt} $
$ \Rightarrow $ $ 6=2\frac{dx}{dt} $ $ \left( \because \frac{dy}{dt}=6given \right) $
$ \Rightarrow $ $ \frac{dx}{dt}=3\,km/h $
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
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