Question

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

• A. $ 120\,\pi\, cm^2/s $
• B. $ 130\,\pi\, cm^2/s $
• C. $ 144\,\pi\, cm^2/s $
• D. $None\, of\, these$

Answer 1

Answer: (C)

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