The side of an equilateral triangle is increasing at the rate of $\displaystyle 3$ cm/s. At what rate its area increasing when the side of the triangle is $\displaystyle 15$ cm ?

Let ' $\displaystyle a$ ' be theside of the triangle, so $\displaystyle \frac{d a}{d t}=3 \mathrm{~cm} / \mathrm{s}$ Now area of an equilateral triangle, $\displaystyle A=\frac{\sqrt{3}}{4} a^{2}$ \[\begin{aligned} & \Rightarrow \frac{d A}{d t}=\frac{\sqrt{3} a}{2} \times \frac{d a}{d t} \\ & \left.\therefore \frac{d A}{d t}\right]_{a=15 \mathrm{~cm}}=\frac{\sqrt{3} \times 15}{2} \times 3=\frac{45 \sqrt{3}}{2} \mathrm{~cm}^{2} / \mathrm{s} \end{aligned} \]