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 \(60π cm^2 /s\).