Question

The sides of a square are increasing at a rate of 10 cm/sec. How fast is the area enclosed by the square increasing when the area is 150 cm2.

Answer 1

$\text{Let x be the side length of the square.}\\[1 em] \therefore \text{ The area of the square is} ~~A=x^{2} \\[1 em] \text{ when the area is} ~150~ \mathrm{cm}^{2}\\[1 em] \therefore \text{The side length is}~x=\sqrt{150} = 5\sqrt{6}~\mathrm{cm} \\[1 em] \therefore x=5\sqrt{6} ~~,~~\frac{d x}{d t}=10\\[1 em] \because A=x^{2} \\[1 em] \begin{aligned} \therefore \frac{d A}{d t}&=2 x \times \frac{d x}{d t} \\[1 em] &=2 \times 5\sqrt{6} \times 10\\[1 em] &=100\sqrt{6}\approx 245 \mathrm{cm}^{2} \mathrm{s} \\[1 em] \end{aligned}\\[1 em] \text{Area is increasing at the rate of}~ 245 ~\mathrm{cm}^{2} / \mathrm{s}$