Question

The diagonal of a square is changing at the rate of $0.5cm/sec$. Then the rate of change of area, when the area is $400c{m}^2$, is equal to

• A. $20\sqrt{2}c{m}^2/sec$
• B. $10\sqrt{2}c{m}^2/sec$
• C. $\frac{1}{10\sqrt{2}}c{m}^2/sec$
• D. $\frac{10}{\sqrt{2}}c{m}^2/sec$

Answer 1

Answer: (B)

Diagonal $D=\sqrt{2}a$
Differentiating w.r.t. t
$\frac{dD}{dt}=\sqrt{2}\frac{da}{at}$
or $\frac{da}{dt}=\frac{1}{\sqrt{2}}\frac{da}{dt}=\frac{1}{\sqrt{2}}\times 0.5cm/s$
Let Area is denoted by $A$
$\frac{dA}{dt}=2a\frac{da}{dt}...\left(i\right)$
when area $A$ is $400c{m}^2$ then $a=20$
$\therefore \frac{dA}{dt}=2\times 20\times \frac{0.5}{\sqrt{2}}=10\sqrt{2}c{m}^2/sec$