The sides of an equilateral triangle are increasing at the rate of 5…

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

Let a be the side. The area of an equilateral triangle is

    \( \displaystyle A = \frac{\sqrt{3}}{4}a^2 \)

Differentiating with respect to time t (chain rule):

    \( \displaystyle \frac{dA}{dt} = \frac{\sqrt{3}}{4}\cdot 2a\,\frac{da}{dt} \)

    \( \displaystyle = \frac{\sqrt{3}}{2}\,a\,\frac{da}{dt} \)

Putting \(a = 20\) and \(\frac{da}{dt} = 5\):

    \( \displaystyle \frac{dA}{dt} = \frac{\sqrt{3}}{2}\times 20 \times 5 \)

    \(= 50\sqrt{3}\,\mathrm{cm^2/sec}\)