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

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

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2cm/sec

1cm/sec

4cm/sec

3cm/sec

answer is D.

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Detailed Solution

$\frac{{dr}_{2}}{dt} = \frac{1}{4} r_{2} = 8 r_{1} = 4 \frac{4}{3} π {r_{2}}^{3} - \frac{4}{3} {πr}_{1}^{3} = k {r_{2}}^{3} - {r_{1}}^{3} = \frac{3 k}{4 π} \Rightarrow 3 {r_{2}}^{2} \frac{{dr}_{2}}{dt} - 3 {r_{1}}^{3} \frac{{dr}_{1}}{dt} = 0 64 \times \frac{1}{4} - 16 \frac{{dr}_{1}}{dt} = 0 \Rightarrow \frac{{dr}_{1}}{dt} = 1$