The ratio of the volume of two cylinders is x : y and the ratio of their diameters is a : b. What is the ratio of their heights ?

Options

A. `xb^(2) : ya^(2)`
B. `xa : yb`
C. xb : ya
D. `xa^(2) : yb^(2)`

Answer

To solve the problem, we need to find the ratio of the heights of two cylinders given the ratio of their volumes and the ratio of their diameters.

### Step-by-Step Solution:

1. **Understand the given ratios**:
- Let the volume of the first cylinder be \( V_1 = x \) and the volume of the second cylinder be \( V_2 = y \).
- The ratio of the volumes is given as \( V_1 : V_2 = x : y \).
- The ratio of their diameters is given as \( d_1 : d_2 = a : b \).

2. **Relate diameters to radii**:
- The radius \( r_1 \) of the first cylinder is \( \frac{d_1}{2} \) and the radius \( r_2 \) of the second cylinder is \( \frac{d_2}{2} \).
- Therefore, the ratio of the radii is \( r_1 : r_2 = \frac{a}{2} : \frac{b}{2} = a : b \).

3. **Volume formula for cylinders**:
- The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
- For the first cylinder:
\[ V_1 = \pi r_1^2 h_1 = \pi (r_1^2) h_1 \]
- For the second cylinder:
\[ V_2 = \pi r_2^2 h_2 = \pi (r_2^2) h_2 \]

4. **Set up the equation using the volumes**:
- From the volumes, we have:
\[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} \]
- This simplifies to:
\[ \frac{x}{y} = \frac{r_1^2 h_1}{r_2^2 h_2} \]

5. **Substituting the radius ratios**:
- Substitute \( r_1 = a \) and \( r_2 = b \):
\[ \frac{x}{y} = \frac{a^2 h_1}{b^2 h_2} \]

6. **Cross-multiply to find the height ratio**:
- Rearranging gives:
\[ x b^2 h_2 = y a^2 h_1 \]
- Thus, we can express the ratio of the heights:
\[ \frac{h_1}{h_2} = \frac{x b^2}{y a^2} \]

7. **Final ratio of heights**:
- Therefore, the ratio of the heights of the two cylinders is:
\[ h_1 : h_2 = x b^2 : y a^2 \]

### Conclusion:
The ratio of the heights of the two cylinders is \( h_1 : h_2 = x b^2 : y a^2 \).