Question

A balloon which always remains spherical is being inflated by pumping in $10$ cubic centimeters of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is $15$ cms.

• A. $\frac {1}{90\pi}cm /sec$
• B. $\frac {1}{9\pi}cm /sec$
• C. $\frac {1}{30\pi}cm /sec$
• D. $\frac {1}{\pi}cm /sec$

Answer 1

Answer: (A)

$\frac {1}{90\pi}cm /sec$

Answer 2

We have, $\frac{d V}{d t} =10 cm ^{3} / s$
$r =15 \,cm$
$\therefore V =\frac{4}{3} \pi r^{3}$
$\Rightarrow \frac{d V}{d r}=4 \pi r^{2}=4 \pi(15)^{2}=900 \pi$
Now, $\frac{d V}{d t} =\frac{d V}{d r} \times \frac{d r}{d t}$
$\Rightarrow 10=900 \pi \times \frac{d r}{d t}$
$\Rightarrow \frac{d r}{d t}=\frac{1}{90 \pi} cm / s$