The radii of the base of a cylinder and a cone are in the ratio `sqrt(3) : sqrt(2)` and their heights are in the ratio `sqrt(2) : sqrt(3)`. Their volumes are in the ratio of

Options

A. `sqrt(3) : sqrt(2)`
B. `3sqrt(3) : sqrt(2)`
C. `sqrt(3) : 2sqrt(2)`
D. `sqrt(2) : sqrt(6)`

Answer

To find the ratio of the volumes of a cylinder and a cone given the ratios of their radii and heights, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the radius of the cylinder be \( r_1 \) and the height be \( h_1 \).
- Let the radius of the cone be \( r_2 \) and the height be \( h_2 \).

2. **Set Up the Ratios:**
- The ratio of the radii is given as:
\[ \frac{r_1}{r_2} = \frac{\sqrt{3}}{\sqrt{2}} \]
- The ratio of the heights is given as:
\[ \frac{h_1}{h_2} = \frac{\sqrt{2}}{\sqrt{3}} \]

3. **Express the Radii and Heights in Terms of a Common Variable:**
- Let \( r_2 = \sqrt{2}y \) and \( r_1 = \sqrt{3}y \) (from the ratio of the radii).
- Let \( h_2 = \sqrt{3}x \) and \( h_1 = \sqrt{2}x \) (from the ratio of the heights).

4. **Write the Volume Formulas:**
- The volume of the cylinder \( V_1 \) is given by:
\[ V_1 = \pi r_1^2 h_1 \]
- The volume of the cone \( V_2 \) is given by:
\[ V_2 = \frac{1}{3} \pi r_2^2 h_2 \]

5. **Substitute the Values:**
- For the cylinder:
\[ V_1 = \pi (\sqrt{3}y)^2 (\sqrt{2}x) = \pi (3y^2)(\sqrt{2}x) = 3\sqrt{2} \pi y^2 x \]
- For the cone:
\[ V_2 = \frac{1}{3} \pi (\sqrt{2}y)^2 (\sqrt{3}x) = \frac{1}{3} \pi (2y^2)(\sqrt{3}x) = \frac{2\sqrt{3}}{3} \pi y^2 x \]

6. **Find the Ratio of the Volumes:**
- Now, we can find the ratio \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{3\sqrt{2} \pi y^2 x}{\frac{2\sqrt{3}}{3} \pi y^2 x} \]
- The \( \pi y^2 x \) terms cancel out:
\[ \frac{V_1}{V_2} = \frac{3\sqrt{2}}{\frac{2\sqrt{3}}{3}} = \frac{3\sqrt{2} \cdot 3}{2\sqrt{3}} = \frac{9\sqrt{2}}{2\sqrt{3}} \]

7. **Simplify the Ratio:**
- To express it in a more standard form, we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ \frac{9\sqrt{2} \cdot \sqrt{3}}{2\sqrt{3} \cdot \sqrt{3}} = \frac{9\sqrt{6}}{6} = \frac{3\sqrt{6}}{2} \]

### Final Answer:
The ratio of the volumes of the cylinder to the cone is:
\[ \frac{3\sqrt{6}}{2} : 1 \]