Question

The edge of a cube is equal to the radius of the sphere. If the rate at which the volume of the cube is increasing is equal to $\lambda$ , then the rate of increase of volume of the sphere

• A. $\frac{4\pi \lambda }{3}$
• B. $4\pi \lambda$
• C. $\frac{\lambda }{3}$
• D. none of these

Answer 1

The correct option is A $\frac{4\pi \lambda }{3}$

Let a be the edge of the cube and r be the radius of the sphere.
Given $a=r$
$\Rightarrow \frac{da}{dt}=\frac{dr}{dt}$
Volume of cube $V=a^3$
$\Rightarrow \frac{dV}{dt}=3a^2\frac{da}{dt}$
$\lambda =3r^2\frac{dr}{dt}$
$\Rightarrow \frac{dr}{dt}=\frac{\lambda }{3r^2}$
Volume of sphere $V^{\prime }=\frac{4}{3}\pi r^3$
$\frac{d{V}^{\prime }}{dt}=4\pi r^2\frac{dr}{dt}$
$\Rightarrow \frac{d{V}^{\prime }}{dt}=\frac{4\pi \lambda }{3}$