The ratio of the volumes of a cone and a cylinder is 8 : 9 and their heights are in the ratio of 2 : 3 What is the ratio of the radii of their bases ?

Options

A. `3 : 4`
B. `3 : 2`
C. `2 : 1`
D. `4 : 3`

Answer

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

### Step-by-Step Solution:

1. **Understand the Volume Formulas**:
- The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r_1^2 h_1 \]
- The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r_2^2 h_2 \]

2. **Set Up the Ratios**:
- We are given that the ratio of the volumes of the cone to the cylinder is:
\[ \frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{8}{9} \]
- Substituting the volume formulas, we have:
\[ \frac{\frac{1}{3} \pi r_1^2 h_1}{\pi r_2^2 h_2} = \frac{8}{9} \]
- Canceling \( \pi \) from both sides:
\[ \frac{r_1^2 h_1}{3 r_2^2 h_2} = \frac{8}{9} \]

3. **Use the Height Ratio**:
- We are also given that the ratio of the heights is:
\[ \frac{h_1}{h_2} = \frac{2}{3} \]
- Therefore, we can express \( h_1 \) in terms of \( h_2 \):
\[ h_1 = \frac{2}{3} h_2 \]

4. **Substitute the Height Ratio**:
- Substitute \( h_1 \) into the volume ratio equation:
\[ \frac{r_1^2 \left(\frac{2}{3} h_2\right)}{3 r_2^2 h_2} = \frac{8}{9} \]
- Simplifying this gives:
\[ \frac{r_1^2 \cdot \frac{2}{3}}{3 r_2^2} = \frac{8}{9} \]
- This simplifies to:
\[ \frac{2 r_1^2}{9 r_2^2} = \frac{8}{9} \]

5. **Cross-Multiply**:
- Cross-multiplying gives:
\[ 2 r_1^2 = 8 r_2^2 \]
- Dividing both sides by 2:
\[ r_1^2 = 4 r_2^2 \]

6. **Find the Ratio of Radii**:
- Taking the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{4} = 2 \]
- Thus, the ratio of the radii of their bases is:
\[ r_1 : r_2 = 2 : 1 \]

### Final Answer:
The ratio of the radii of their bases is \( 2 : 1 \).