Question

A man, $1.6m$ tall, walks away from a lamp which is $4m$ above ground, at the rate of $30m/min$.If his shadow increases at the rate of $\lambda m/min$, then the value of $\lambda$ is

Answer 1

Let $PQ=4m$ be the height of lamp and $AB=1.6m$ be height of man. Let the end of shadow is $R$ and it is at a distance of $I$ from $A$ when the man is at a distance of $x$ from $PQ$, at some instance.
Since $△PQR$ and $△ABR$ are similar,Thus
$\frac{PQ}{AB}=\frac{PR}{AR}$
$\Rightarrow \frac{4}{1.6}=\frac{x+l}{l}$
$\Rightarrow 2x=3l$
$\Rightarrow 2\frac{dx}{dt}=3\frac{dl}{dt}$
$\{\because \frac{dx}{dt}=30m/min\}$
$\therefore \frac{dl}{dt}=\frac{2}{3}\cdot 30m/min=20m/min$(lengthening)
Thus $\lambda =20$