Question

The radius and height of a cylinder are equal. If the radius of the sphere is equal to the height of the cylinder, then the ratio of the rates of increase of the volume of the sphere and the volume of the cylinder is

• A. $4:3$
• B. $3:4$
• C. $4:3\pi$
• D. $3:4\pi$

Answer 1

The correct option is A $4:3$

Volume of cylinder $V_1=\pi r^{2}h$

$\Rightarrow V_1=\pi h^3$

$\frac{d{V}_1}{dt}=3\pi h^2\frac{dh}{dt}$

Volume of sphere $V_2=\frac{4}{3}\pi R^3$

$\Rightarrow \frac{d{V}_2}{dt}=4\pi R^2\frac{dR}{dt}$

Given $h=R$

$\Rightarrow \frac{dh}{dt}=\frac{dR}{dt}$

So, $\frac{\frac{d{V}_2}{dt}}{\frac{d{V}_1}{dt}}=\frac{4}{3}$