Question

If the rate of change in the circumference of a circle is $0.5\text{cm/sec}$, then the rate of change in its area, when the circumference is $10\pi \text{cm},$ is equal to(in $\text{sq.cm/sec}$)

• A. $2.25$
• B. $2.1$
• C. $3.1$
• D. $2.5$

Answer 1

The correct option is D $2.5$

Circumference of a circle, $P=2\pi r\left(r\text{is the radius}\right)$
Given, $\frac{dP}{dt}=0.5\text{cm/sec},P=10\pi \text{cm}$
$\Rightarrow 2\pi r=10\pi \Rightarrow r=5\text{cm}$
Now, $\frac{dP}{dt}=2\pi \cdot \frac{dr}{dt}$
$\Rightarrow 0.5=2\pi \cdot \frac{dr}{dt}$
$\therefore \frac{dr}{dt}=\frac{1}{4\pi }\text{cm/sec}$
Now area of circle $A=\pi r^2$
$\Rightarrow \frac{dA}{dt}=\pi \cdot 2r\cdot \frac{dr}{dt}=2.5\text{sq.cm/sec}$