Question

If the diagonal of a square is changing at the rate of $0.5cm/s$.

Then, the rate of change of area, when the area is $400c{m}^2$, is

• A. $20\sqrt{2}c{m}^2/s$
• B. $10\sqrt{2}c{m}^2/s$
• C. $\frac{1}{10\sqrt{2}}c{m}^2/s$
• D. $\frac{10}{\sqrt{2}}c{m}^2/s$
• E. $5\sqrt{2}c{m}^2/s$

Answer 1

The correct option is B

$10\sqrt{2}c{m}^2/s$

Explanation for the correct option:

Step-1: Determine the rate of change on the side

Let the side length of the square is $acm$.

Thus, the length of the diagonal is $s=a\sqrt{2}cm$.

Let the time denoted by $t$.

Differentiate both sides of the equation with respect to $t$.

$\frac{ds}{dt}=\frac{da}{dt}\sqrt{2}$.

It is given that the rate of change in diagonal is $\frac{ds}{dt}=0.5cm/s$.

Thus, $\frac{da}{dt}\sqrt{2}=0.5cm/s$

$\Rightarrow \frac{da}{dt}=\frac{1}{2\sqrt{2}}cm/s$

Step-2: Determine the rate of change in the area

The area of the square is $A=a^2$.

Differentiate both sides of the equation with respect to $t$.

$\frac{dA}{dt}=2a\frac{da}{dt}$.

Substitute $\frac{da}{dt}=\frac{1}{2\sqrt{2}}cm/s$ in the above equation.

$$\begin{gathered} \Rightarrow \frac{dA}{dt}=2a\left(\frac{1}{2\sqrt{2}}\right) \\ \Rightarrow \frac{dA}{dt}=\frac{a}{\sqrt{2}}c{m}^2/s \end{gathered}$$

The given area is $A=400c{m}^2$.

$$\begin{gathered} \Rightarrow a^2={\left(20cm\right)}^2 \\ \Rightarrow a=20cm \end{gathered}$$

Therefore, $\frac{dA}{dt}=\frac{20}{\sqrt{2}}c{m}^2/s$

$$\begin{gathered} \Rightarrow \frac{dA}{dt}=\frac{20}{\sqrt{2}} \\ \Rightarrow \frac{dA}{dt}=\frac{10\times 2}{\sqrt{2}} \\ \Rightarrow \frac{dA}{dt}=10\sqrt{2}c{m}^2/s \end{gathered}$$

Therefore, the rate of change of area, when the area is $400c{m}^2$, is $10\sqrt{2}c{m}^2/s$.

Hence, option$\left(B\right)$ i.e. $10\sqrt{2}c{m}^2/s$ is the correct option.