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

Explanation

`V=frac{4}{3} pi(r_{1}^{3}-r_{2}^{3})`


`Rightarrow frac{d V}{d t}=4 pi(r_{1}^{2} frac{d r_{1}}{d t}-r_{2}^{2} frac{d r_{2}}{d t})`


`Rightarrow r_{1}^{2} frac{d r_{1}}{d t}=r_{2}^{2} frac{d r_{2}}{d t}[because frac{d V}{d t}=0]`


`Rightarrow(4)^{2} times 1=(8)^{2} frac{d r_{2}}{d t}`


`Rightarrow frac{d r_{2}}{d t}=frac{1}{4} {cm} / {sec}`