A boat 10 m high floating at a uniform speed of 13 meters per minute (m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is:

26 m/min

The correct answer is Option (1) → 26 m/min

Let the distance between the lamp post and the boat be $x$ m, and the length of the boat’s shadow be $y$ m.

Height of lamp post = 15 m, height of boat = 10 m.

By similarity of triangles:

$\frac{15}{x + y} = \frac{10}{y}$

$15y = 10(x + y)$

$15y = 10x + 10y$

$5y = 10x$

$y = 2x$

Differentiate w.r.t. time $t$:

$\frac{dy}{dt} = 2\frac{dx}{dt}$

Given $\frac{dx}{dt} = 13$ m/min

$\Rightarrow \frac{dy}{dt} = 2(13) = 26$ m/min

Rate of increase of shadow = 26 m/min