Question

If the length of the diagonal of a square is increasing at the rate of 0.2 cm/sec, then rate of increase of its area when its side is $30/\sqrt{2}$ cm, is

• A. $3$ cm${}^2$/sec
• B. $\frac{6}{\sqrt{2}}$ cm${}^2$/sec
• C. $\sqrt[3]{32}$ cm${}^2$/sec
• D. $6$ cm${}^2$/sec

Answer 1

Answer: (D)

$6$ cm${}^2$/sec

Length of diagonal of a square $l=x\sqrt{2}$ where $x$ is a side of square.

$\Rightarrow \frac{dl}{dt}=\sqrt{2}\frac{dx}{dt}$
$\Rightarrow \frac{dx}{dt}=\frac{0.2}{\sqrt{2}}$

Area of square $A=x^2$
$\frac{dA}{dt}=2x\frac{dx}{dt}$
$\Rightarrow \frac{dA}{dt}=2.\frac{30}{\sqrt{2}}.\frac{0.2}{\sqrt{2}}=6c{m}^2/sec$