Question

If the rate of change in the circumference of a circle of $0.3cm/s$, then the rate of change in the area of the circle when the radius is $5$cm, is:

• A. $1.5$sq cm/s
• B. $0.5$ sq cm/s
• C. $5$ sq cm/s
• D. $3$ sq cm/s

Answer 1

The correct option is A $1.5$sq cm/s

Let $C$ be the circumference, $A$ be the area and $r$ be the radius of the circle.

Thus,

$A=\pi r^2$ and $C=2\pi r$
and $\frac{dC}{dt}=0.3$

Now,

$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$ and $\frac{dC}{dt}=2\pi \frac{dr}{dt}$

$\Rightarrow \frac{dr}{dt}=\frac{0.3}{2\pi }$cm/s

$\therefore {\frac{dA}{dt}|}_{r=5}=2\pi \cdot 5\cdot \frac{0.3}{2\pi }=1.5$sq cm/s