Question

The radius of a circle is uniformly increasing at the rate of $3cm/s$. What is the rate of increase in area, when the radius is $10cm$?

• A. $6\pi$ ${cm}^2/s$
• B. $10\pi$ ${cm}^2/s$
• C. $30\pi$ ${cm}^2/s$
• D. $60\pi$ ${cm}^2/s$

Answer 1

The correct option is D $60\pi$ ${cm}^2/s$

Let $r$ be the radius of the circle and $A$ be its area.
Then, $\frac{dr}{dt}=3cm/s$

Now, $A=\pi r^2$

$\Rightarrow \frac{dA}{dt}=\frac{d\left(\pi r^2\right)}{dt}=2\pi r\frac{dr}{dt}=2\pi \left(10cm\right)\left(3cm/s\right)=60\pi {cm}^2/s$
Option D is the correct answer.