Question

The length of the side of a square sheet of metal is increasing at the rate of $4cm/sec$. The rate at which the area of the sheet is increasing when the length of its side is $8cm$, is

• A. $16c{m}^2/sec$
• B. $8c{m}^2/sec$
• C. $64c{m}^2/sec$
• D. None of these

Answer 1

The correct option is C

$64c{m}^2/sec$

Find the rate at which the area of the square sheet is increasing

Let the side of the square be $xcm$

The area of the square $\left(A\right)=x^{2}c{m}^2$

Differentiating $A$ with respect to time $\left(t\right)$

$\frac{dA}{dt}=\frac{d}{dt}\left(x^2\right)$

$=2x\frac{dx}{dt}$

$\therefore$ $\frac{dA}{dt}=8x$ $...\left(\because \frac{dx}{dt}=4cm/sec\right)$

$\therefore$ ${\frac{dA}{dt}}_{\left|x=8\right|}=8\times 8=64c{m}^2/sec$

So, the area is increasing at a rate of $64c{m}^2/sec$ when the side of the square is $8cm$.

Hence option $\left(C\right)$ is the correct answer.