Question

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing, is:

• A. a constant
• B. proportional to the radius
• C. inversely proportional to the radius
• D. inversely proportional to the surface area
• E. proportional to its surface area

Answer 1

Answer: (D)

Given that, $\frac{dV}{dt}=k$ (say) ...(i)
$\because$ $V=\frac{4}{3}\pi R^3$
On differentiating w.r.t. t, we get
$\frac{dV}{dt}=4\pi R^2\frac{dR}{dt}$
$\Rightarrow$ $\frac{dR}{dt}=\frac{k}{4\pi R^2}$ [from(i)]
$\Rightarrow$ Rate of increasing radius is inversely proportional to its surface area.