Multiple choice

The volume of metallic hallow sphere is constant. If the outer radius is increasing at the rate of $V\ cm/sec$. Then the rate at which the inner radius increasing when the radii are $a + d, a$ is

  1. $\dfrac {V(a + d)^{2}}{a^{2}}$
  2. $\dfrac {V(a + d)}{a}$
  3. $V(a + d)$
  4. $a + d$
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A Correct answer
Explanation

For a hollow sphere with constant metallic volume, R^3 - r^3 is constant. Differentiating gives R^2 dR/dt = r^2 dr/dt, so with R = a + d, r = a, and dR/dt = V, the inner-radius rate is V(a + d)^2/a^2.

AI explanation

The volume of the hollow sphere is constant, given by V = (4/3)(pi)((a+d)^3 - a^3). Differentiating with respect to time gives 0 = (a+d)^2(da/dt) - a^2(da/dt for inner). If the outer radius increases at rate V, then 0 = V(a+d)^2 - a^2(v). Solving for the rate of increase of the inner radius v gives v = V(a + d)^2 / a^2.

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