Question

From the given diagram, find the ratio of the volume of the sphere to the volume of the cylinder.

Answer 1

We know that the volume of the sphere is given by the equation:
Volume of the sphere $=\frac{4}{3}\pi r^3$
As per the diagram,
diameter of the sphere $=d=2R$
$\Rightarrow$ Radius of the sphere $=r=\frac{d}{2}=\frac{2R}{2}=R$
Volume of the sphere $\left(V_s\right)=\frac{4}{3}\pi R^3$……..(1) (substituting for r)

Also, the volume of the cylinder is given by the equation:
Volume of the cylinder $=\pi r^{2}h$

As per the diagram,
radius of the cylinder $=r=R$
Height of the cylinder $=h=2R$
Volume of the cylinder$\left(V_c\right)=\pi r^{2}h$
$=\pi R^2\left(2R\right)$ (substituting for r and h)
$=2\pi R^3$………(2)

Taking the ratio (1) and (2)
$\frac{\text{Volume of the sphere (Vs)}}{\text{Volume of the cone (Vc)}}=\frac{\frac{4}{3}\pi R^3}{2\pi R^3}=\frac{2}{3}$

Therefore, the ratio of the volume of the sphere to the volume of the cylinder is $2:3$.
$\to$ Option D is correct.