Question

The volume of metal in a hollow sphere is constant.If the inner radius is increasing at the rate of $1cm/sec$, then the rate of increase of the outer radius when the radii are $4cm$ and $8cm$ respectively is.

• A. $0.75cm/sec$
• B. $0.25cm/sec$
• C. $1cm/sec$
• D. $0.50cm/sec$

Answer 1

The correct option is B $0.25cm/sec$

$V=\frac{4}{3}\pi R^3-\frac{4}{3}\pi r^3$
volume is constant.
$\Rightarrow \frac{dV}{dt}=0$
$\Rightarrow 4\pi R^2\frac{dR}{dt}-4\pi r^2\frac{dr}{dt}=0$
$\Rightarrow 64\frac{dR}{dt}-16\left(1\right)=0$
$\therefore \frac{dR}{dt}=0.25cm/sec$
Hence, option B.