Question

The radius of a circle is increasing uniformly at the rate of $3 cm/s$. Find the rate at which the area of the circle is increasing when the radius is $10 cm$.

Answer 1

The correct answer is $60π cm^2 /s$
The area of a circle $(A)$ with radius $(r)$ is given by,
$A=πr^2$
Now, the rate of change of area $(A)$ with respect to time $(t)$ is given by,
$\frac{dA}{dt}=\frac{d}{dt}(πr^2).\frac{dr}{dt}=2πr\frac{dr}{dt} ....$[By chain rule]
It is given that,
$\frac{dr}{dt}=3 cm/s$
$∴ \frac{dA}{dt}=2πr(3)=6πr$
Thus, when $r = 10 cm,$
$\frac{dA}{dt}=6π(10)=60π cm^2/s$
Hence, the rate at which the area of the circle is increasing when the radius is $10 cm$ is $60π cm^2 /s$.