Question

A spherical iron ball$10$ cm in radius is with a layer of ice of uniform thickness that melts at a rate of $50$ cm${}^3$/min. When the thickness of ice is $5$ cm, then the rate at which the thickness of ice decreases, is

• A. $\frac{5}{6\pi }$ cm/min
• B. $\frac{1}{54\pi }$ cm/min
• C. $\frac{1}{18\pi }$ cm/min
• D. $\frac{1}{36\pi }$ cm/min

Answer 1

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

Let the thickness of ice be $x$ cm at any instant.

$r_1-r_2=x$

$\Rightarrow r_1=10+x$

Volume $V=\frac{4}{3}\pi (10+x{)}^3$

$\Rightarrow \frac{dV}{dt}=4\pi (10+5{)}^2\frac{dx}{dt}$

$\Rightarrow 50=4\pi \left(225\right)\frac{dx}{dt}$

$\Rightarrow \frac{dx}{dt}=\frac{1}{18\pi }cm/minute$