Question

Side of an equilateral triangle expands at the rate of 2 cm/sec. The rate of increase of its area when each side is 10 cm is
(a) $10\sqrt{2}{\mathrm{cm}}^2/\sec$
(b) $10\sqrt{3}{\mathrm{cm}}^2/\sec$
(c) 10 cm²/sec
(d) 5 cm²/sec

Answer 1

(b) $10\sqrt{3}{\mathrm{cm}}^2/\sec$

$$\begin{gathered} \text{Let}\mathit{\text{x}}\text{be the side and}\mathit{\text{A}}\text{be the area of the equilateral triangle at any time}\mathit{\text{t.}}\text{Then,} \\ A=\frac{\sqrt{3}}{4}{\mathit{\text{x}}}^2 \\ \Rightarrow \frac{dA}{dt}=2\times \frac{\sqrt{3}}{4}{\mathit{\text{x}}}^{}\frac{dx}{dt} \\ \text{⇒}\frac{dA}{dt}\text{=}\frac{\sqrt{3}}{2}\text{×2×10} \\ \text{⇒}\frac{dA}{dt}\text{=10}\sqrt{3}{\text{cm}}^2\text{/sec} \end{gathered}$$