Question:
The side of an equilateral triangle is increasing at the rate of $\frac{1}{3} \mathrm{~cm} / \mathrm{sec}$. Find the rate of increase of its perimeter.
Solution:
Let x be the side and P be the perimeter of the equilateral triangle at any time t. Then,
$P=3 x$
$\Rightarrow \frac{d P}{d t}=3 \frac{d x}{d t}$
$\Rightarrow \frac{d P}{d t}=3 \times \frac{1}{3}$ $\left[\because \frac{d x}{d t}=\frac{1}{3} \mathrm{~cm} / \mathrm{sec}\right]$
$\Rightarrow \frac{d P}{d t}=1 \mathrm{~cm} / \mathrm{sec}$