The radii of the base of a cylinder and a cone are in the ratio of 3 : 4 and their heights are in the ratio 2 : 3. Then the ratio of their volumes is :

Question

Accepted Answer

To determine the ratio of the volumes of the cylinder and the cone, we need to use the formulas for the volumes of these shapes and apply the given ratios for their dimensions.

1. **Volume of a Cylinder**: The formula for the volume of a cylinder is $ V_{\text{cylinder}} = \pi r^2 h $, where $ r $ is the radius of the base and $ h $ is the height.

2. **Volume of a Cone**: The formula for the volume of a cone is $ V_{\text{cone}} = \frac{1}{3} \pi r^2 h $, where $ r $ is the radius of the base and $ h $ is the height.

Given:
- The ratio of the radii of the base of the cylinder to the cone is 3:4.
- The ratio of their heights is 2:3.

Let's denote:
- $ r_{\text{cylinder}} = 3x $ and $ r_{\text{cone}} = 4x $.
- $ h_{\text{cylinder}} = 2y $ and $ h_{\text{cone}} = 3y $.

**Step-by-step Calculation**:

1. **Volume of the Cylinder**:
   $$
   V_{\text{cylinder}} = \pi (3x)^2 (2y) = \pi \times 9x^2 \times 2y = 18\pi x^2 y
   $$

2. **Volume of the Cone**:
   $$
   V_{\text{cone}} = \frac{1}{3} \pi (4x)^2 (3y) = \frac{1}{3} \pi \times 16x^2 \times 3y = 16\pi x^2 y
   $$

3. **Ratio of Volumes**:
   $$
   \text{Ratio} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{18\pi x^2 y}{16\pi x^2 y} = \frac{18}{16} = \frac{9}{8}
   $$

Thus, the ratio of the volumes of the cylinder to the cone is 9:8, which corresponds to Option 2.

**Explanation of Incorrect Options**:

- **Option 1 (1:2)**: This ratio does not account for the specific calculations involving the squared radii and the heights. The derived ratio from the calculations is 9:8, not 1:2.

- **Option 3 (3:4)**: This ratio might seem plausible if one only considers the ratio of the radii or heights individually, but it does not reflect the combined effect of both dimensions on the volumes.

- **Option 4 (4:3)**: This ratio is incorrect because it does not match the calculated ratio of the volumes. The calculations clearly show that the ratio is 9:8.

In conclusion, the correct ratio of the volumes of the cylinder and the cone, based on the given ratios of their dimensions, is 9:8, which is Option 2.