The diameters of two cylinders are in the ratio 3:2 and their volumes are equal. The ratio of their heights is

Options

A. `2:3`
B. `3:2`
C. `9:4`
D. `4:9`

Answer

To solve the problem, we need to find the ratio of the heights of two cylinders given that their diameters are in the ratio of 3:2 and their volumes are equal.

### Step-by-Step Solution:

1. **Define the Diameters and Radii**:
- Let the diameter of the first cylinder (D1) be \(3x\).
- Therefore, the radius of the first cylinder (r1) is \( \frac{3x}{2} \).
- Let the diameter of the second cylinder (D2) be \(2x\).
- Therefore, the radius of the second cylinder (r2) is \( \frac{2x}{2} = x \).

2. **Volume of the Cylinders**:
- The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
- For the first cylinder, the volume (V1) can be expressed as:
\[ V_1 = \pi \left(\frac{3x}{2}\right)^2 h_1 = \pi \left(\frac{9x^2}{4}\right) h_1 \]
- For the second cylinder, the volume (V2) can be expressed as:
\[ V_2 = \pi (x^2) h_2 \]

3. **Set the Volumes Equal**:
- Since the volumes are equal, we set \(V_1\) equal to \(V_2\):
\[ \pi \left(\frac{9x^2}{4}\right) h_1 = \pi (x^2) h_2 \]

4. **Cancel Out Common Terms**:
- We can cancel \(\pi\) from both sides:
\[ \frac{9x^2}{4} h_1 = x^2 h_2 \]
- Next, we can cancel \(x^2\) from both sides (assuming \(x \neq 0\)):
\[ \frac{9}{4} h_1 = h_2 \]

5. **Rearranging for Height Ratio**:
- Rearranging gives us:
\[ h_1 = \frac{4}{9} h_2 \]
- To find the ratio of the heights \(h_1:h_2\), we can express it as:
\[ \frac{h_1}{h_2} = \frac{4}{9} \]

6. **Final Ratio**:
- Thus, the ratio of the heights \(h_1:h_2\) is \(4:9\).

### Conclusion:
The ratio of the heights of the two cylinders is \(4:9\).