Question

The height of a right circular cylinder is equal to Its diameter. If it is melted and recast into a sphere of radius equal to the radius of the cylinder, find the part of the material that remained unused

• A. $\frac{1}{3}$ rd of volume of cylinder
• B. $\frac{1}{2}$ of volume of cylinder
• C. equal to volume of cylinder
• D. $0$

Answer 1

The correct option is A $\frac{1}{3}$ rd of volume of cylinder

Let h be height of the cylinder. Then, its diameter is h and so its radius is $\frac{h}{2}$. Hence, its volume is

$V_1=\pi {\left(\frac{h}{2}\right)}^2\cdot h=\frac{\pi h^2}{4}$

Again, the radius of the sphere $=\frac{h}{2}$

Hence, the volume of the sphere is

$V_2=\frac{4}{3}\pi {\left(\frac{h}{2}\right)}^3=\frac{\pi h^3}{6}$

$\therefore$ The volume of the unused material $=V_1-V_2$

$=\frac{\pi h^3}{4}-\frac{\pi h^3}{6}=\frac{\pi h^3(3-2)}{12}=\frac{\pi h^3}{12}=\frac{1}{3}\cdot \frac{\pi h^3}{4}=\frac{1}{3}{V}_1$

Hence, the required volume of the unused material is equal to $\frac{1}{3}rd$ of the volume of the cylinder.