The sides of an equilateral triangle are increasing at the rate of 5 cm/sec. The rate at which the area increases when the side is 20 cm, is

$50\sqrt{3}\, cm^2/sec$

The correct answer is Option (2) → $50\sqrt{3}\, cm^2/sec$

$\text{Let side of the equilateral triangle be }a$

$\text{Area }A=\frac{\sqrt{3}}{4}a^{2}$

Given: $\frac{da}{dt}=5\text{ cm/s}$

$\frac{dA}{dt}=\frac{\sqrt{3}}{4}\times2a\frac{da}{dt}=\frac{\sqrt{3}}{2}a\frac{da}{dt}$

When $a=20$:

$\frac{dA}{dt}=\frac{\sqrt{3}}{2}\times20\times5=50\sqrt{3}$

$\frac{dA}{dt}=50\sqrt{3}\ \text{cm}^2/\text{s}$