Question

The rate of change of the area of a circle with respect to its radius $r$ at

• A. $r=3cm$ is $6{\pi }^{2}c{m}^2/cm$
• B. $r=4cm$ is $4\pi c{m}^2/cm$
• C. Both (a) and (b) are correct
• D. Both (a) and (b) are incorrect

Answer 1

Answer: (D)

Let $A$ denotes the area of the circle when its radius is $r$, then $A=\pi r^2$.
Now, the rate of change of the area with respect to its radius is given by,
$\frac{dA}{dr}=\frac{d}{dr}\left(\pi r^2\right)=2\pi r$
(a) When $r=3cm,{\left(\frac{dA}{dr}\right)}_{r=3}=2\pi (3)=6\pi c{m}^2/cm$.
Hence, the area of the circle is changing at the rate of $6\pi c{m}^2/cm$ when its radius is $3cm$.
(b) When $r=4cm,{\left(\frac{dA}{dt}\right)}_{r=4}=2\pi (4)=8\pi c{m}^2$ per $cm$.
Hence, the area of the circle is changing at the rate of $8\pi c{m}^2/cm$ when its radius is $4cm$.