Question

A spherical iron ball of radius $10cm$, coated with a layer of ice of uniform thickness, melts at a rate of $100\pi cubiccm/min$. The rate at which the thickness layer decreases, when the thickness of ice is $5cm$, is

• A. $1cm/min$
• B. $2cm/min$
• C. $\frac{1}{376}cm/min$
• D. $5cm/min$
• E. $3cm/min$

Answer 1

The correct option is A

$1cm/min$

Explanation for the correct option:

Step 1. Put the value of $V$ in $\frac{dV}{dt}$:

$\frac{dV}{dt}=100\mathrm{\pi }c{m}^3/min$

$\Rightarrow \frac{d}{dt}\left(\frac{4}{3}{\mathrm{\pi r}}^3\right)=100\pi$

$\Rightarrow 3r^2\left(\frac{dr}{dt}\right)=\frac{300\pi }{4\pi }$

Step 2. Put the value of $r=5$:

$\Rightarrow \left(\frac{dr}{dt}\right)=\frac{300}{4\times 3\times 25}$

$=1cm/min$

$\therefore$The rate of decrease of thickness is $\mathbf{1}\mathbf{}\mathit{c}\mathit{m}\mathbf{/}\mathit{m}\mathit{i}\mathit{n}$

Hence, Option ‘A’ is Correct.