Question

A right circular cone, a right circular cylinder and a hemisphere, all have the same radius, and the heights of the cone and cylinder are equal to their diameters. Then their volumes are proportional, respectively, to:

• A. $1 : 3 : 1$
• B. $2 : 1 : 3$
• C. $3 : 2 : 1$
• D. $1 : 2 : 3$

Hint:

When comparing volumes, factor out common terms like $\pi r^3$ before simplifying the ratio to avoid mistakes.

Answer 1

The Correct Option is A

Let the common radius be $r$.
For the cone: Height $h = 2r$. Volume of cone = $\frac{1}{3} \pi r^2 h = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3$.
For the cylinder: Height $h = 2r$. Volume of cylinder = $\pi r^2 h = \pi r^2 (2r) = 2\pi r^3$.
For the hemisphere: Volume of hemisphere = $\frac{2}{3} \pi r^3$.
Now, let's write their volumes in ratio form:
Cone : Cylinder : Hemisphere = $\frac{2}{3} \pi r^3 : 2\pi r^3 : \frac{2}{3} \pi r^3$.
Cancel $\frac{2}{3} \pi r^3$ from each term: $1 : 3 : 1$.
Thus, the ratio is $\boxed{1 : 3 : 1}$.