Question

A spherical ball of radius $10$cm is enclosed by ice of uniform thickness in spherical shape. If ice melts at the rate of $50c{m}^3/min$, then the rate of decrease of thickness of ice when thickness of ice is $5$cm is?

• A. $\frac{1}{36\pi }cm/min$
• B. $\frac{1}{9\pi }cm/min$
• C. $\frac{1}{18\pi }cm/min$
• D. $\frac{2}{9\pi }cm/min$

Answer 1

The correct option is C $\frac{1}{18\pi }cm/min$

$r=10$cm
volume of ice $=\frac{4}{3}\pi (r+x{)}^3-\frac{4}{3}\pi r^3$
$\frac{dv}{dt}=50c{m}^3/min$
$4\pi (r+x{)}^2\frac{dx}{dt}=50$
$4\pi (15{)}^2\frac{dx}{dt}=50$ at $r=10$ and $x=5$
$\frac{dx}{dt}=\frac{50}{4\pi \left(225\right)}=\frac{1}{18\pi }cm/min$.