The area of a circle of radius r increases at the rate of $5\, cm^2/sec$; find the rate at which the radius increases. Also find the value of this rate when the circumference is 10 cm.

$\frac{1}{2}\,cm/s$

The correct answer is Option (1) → $\frac{1}{2}\,cm/s$

Let r cm be the radius and A be the area enclosed by it at any time t seconds, then

$Α = πr^2$ ...(i)

Differentiating (i) w.r.t. t, we get

$\frac{dA}{dr}= π.2r\frac{dr}{dt}=2πr\frac{dr}{dt}$ but $\frac{dA}{dt}= 5\, cm^2/sec$ (given)

$⇒ 5 = 2πг\frac{dr}{dt}⇒\frac{dr}{dt}=\frac{5}{2πr}$

Hence, the radius is increasing at the rate of $\frac{5}{2πr}\,cm/sec$.

When the circumference is 10 cm i.e. $2πг=10 cm$,

Then $\frac{dr}{dt}=\frac{5}{10}\,cm/sec = \frac{1}{2}\, cm/sec$.