Question

A stone is dropped into a quiet lake and waves move in circles at the speed of $6cm$ per second. At the instant, when the radius of the circular wave is $12cm$, the enclosed area is increasing at the rate of

• A. $120\pi c{m}^2/s$
• B. $130\pi c{m}^2/s$
• C. $144\pi c{m}^2/s$
• D. $Noneofthese$

Answer 1

Answer: (C)

Given that,
Velocity, $\frac{dr}{dt}=6cm/s$
Also, given radius $r=12cm$
Area = Area of circle (waves)
$A=\pi r^2$
On differentiating w.r.t. t
$\frac{dA}{dt}=\frac{d}{dt}\left(\pi r^2\right)$
$=\frac{d}{dr}\left(\pi r^2\right)\left(\frac{dr}{dt}\right)$
$\frac{dA}{dt}=2\pi r\left(\frac{dr}{dt}\right)$
$\frac{dA}{dt}=2\left(\pi \right)\left(12\right)\left(6\right)$
$\frac{dA}{dt}=144\pi c{m}^2/s$
Increasing area $=\frac{dA}{dt}$
$=144\pi c{m}^2/s$