Question:
Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is
(a) $8 \sqrt{3} \mathrm{~cm}^{2} / \mathrm{hr}$
(b) $4 \sqrt{3} \mathrm{~cm}^{2} / \mathrm{hr}$
(c) $\frac{\sqrt{3}}{8} \mathrm{~cm}^{2} / \mathrm{hr}$
(d) none of these
Solution:
(a) $8 \sqrt{3} \mathrm{~cm}^{2} / \mathrm{hr}$
Let $x$ be the side and $A$ be the area of the equilateral triangle at any time $t$. Then,
$A=\frac{\sqrt{3}}{4} x^{2}$
$\Rightarrow \frac{d A}{d t}=\frac{\sqrt{3}}{2} x\left(\frac{d x}{d t}\right)$
$\Rightarrow \frac{d A}{d t}=\frac{\sqrt{3}}{2}(2)(8)$
$\Rightarrow \frac{d A}{d t}=8 \sqrt{3} \mathrm{~cm}^{2} / \mathrm{hr}$