The median of an equilateral triangle is increasing at the rate of $2\sqrt{3} \text{ cm/s}$. Find the rate at which its side is increasing.

$4\text{ cm/s}$

The correct answer is Option (3) → $4\text{ cm/s}$ ##

Let the length of the median be $x \text{ cm}$.

And side of equilateral triangle be $y \text{ cm}$.

In $\triangle ABD$, $AB^2 = AD^2 + BD^2 \ (\angle D = 90^{\circ})$

$y^2 = x^2 + \left( \frac{y}{2} \right)^2$

$\frac{3}{4}y^2 = x^2$

$y^2 = \frac{4}{3}x^2$

$y = \frac{2}{\sqrt{3}}x$

$\frac{d}{dt}(y) = \frac{2}{\sqrt{3}} \frac{d}{dt}(x)$

$\frac{dy}{dt} = \frac{2}{\sqrt{3}} \frac{dx}{dt}$

$\frac{dy}{dt} = \frac{2}{\sqrt{3}} \times 2\sqrt{3}$

$= 4 \text{ cm/s}$

Hence, side of equilateral triangle increases at $4 \text{ cm/s}$.