Question

A football is inflated by pumping air in it. When it acquires spherical shape its radius increases at the rate of $0.02\, cm/s$. The rate of increase of its volume when the radius is $10\, cm$ is ___________$\pi$ cm/s

• A. 0
• B. 2
• C. 8
• D. 9

Answer 1

Answer: (C)

$v = \frac{4}{3}\pi r^{3}$
$\frac{dv}{dt} = \frac{4}{3}\pi \frac{d}{dt}r^{3} = \frac{4}{3}\pi 3r^{2}. \frac{dr}{dt} = 4\pi r^{2}.\frac{dr}{dt}$
when $r = 10 \,cm;$
$\frac{dv}{dt} = 4\pi\left(10\right)^{2}.\left(0.02\right)= 8\, \pi\, cm^{3}/s.$