Question

If the ratio of base radii of right circular cylinder and cone is 2 : 3 and the ratio of their heights is 3: 4, then the ratio of their volumes is

• A. 1:1
• B. 4:1
• C. 9:8
• D. 1:3

Answer 1

The Correct Option is A

To solve the problem, we need to find the ratio of volumes of a right circular cylinder and a cone given the ratios of their radii and heights.

1. Volume Formulas:

The volume of a cylinder is given by:

$ V_{\text{cylinder}} = \pi r_1^2 h_1 $

The volume of a cone is given by:

$ V_{\text{cone}} = \frac{1}{3} \pi r_2^2 h_2 $

2. Given Ratios:

Ratio of radii: $r_1 : r_2 = 2 : 3$

Ratio of heights: $h_1 : h_2 = 3 : 4$

3. Expressing Volumes in Terms of Ratios:

Let $r_1 = 2$, $r_2 = 3$, $h_1 = 3$, $h_2 = 4$

Then:
$V_{\text{cylinder}} = \pi (2)^2 (3) = \pi \cdot 4 \cdot 3 = 12\pi$

$V_{\text{cone}} = \frac{1}{3} \pi (3)^2 (4) = \frac{1}{3} \pi \cdot 9 \cdot 4 = 12\pi$

4. Finding the Ratio:
$ \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{12\pi}{12\pi} = 1 : 1 $

Final Answer:
The ratio of their volumes is $1 : 1$