Question

A cylinder and a cone have the same base radius and the same volume. What's the ratio of the height of the cylinder to the height of the cone?

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

Answer 1

Answer:

A. 1 : 3

To find the ratio of the height of the cylinder to the height of the cone, let's set up the formulas for their volumes.

Given Data:

• Let the base radius of both the cylinder and the cone be $r$.

• Let the height of the cylinder be $h_1$.

• Let the height of the cone be $h_2$.

• Their volumes are equal.

Step-by-Step Calculation:

1. Write the volume formulas:

   
   

   • Volume of the cylinder ($V_1$) = $\pi r^2 h_1$

   • Volume of the cone ($V_2$) = $\frac{1}{3}\pi r^2 h_2$

2. Equate the two volumes (since they are equal):

   
   

   $\pi r^2 h_1 = \frac{1}{3}\pi r^2 h_2$

   Simplify the equation: Divide both sides by $\pi r^2$:

   
   

   $h_1 = \frac{1}{3} h_2$

3. Find the ratio ($h_1 : h_2$):

   
   

   $\frac{h_1}{h_2} = \frac{1}{3}$

Answer:

The ratio of the height of the cylinder to the height of the cone is $1:3$.