The diameter of two cylinders, whose volumes are equal, are in the ratio 3:2. Their heights will be in the ratio

Options

A. `4:9`
B. `5:6`
C. `5:8`
D. `8:9`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We are given that the diameters of two cylinders are in the ratio 3:2, and their volumes are equal. We need to find the ratio of their heights.

### Step 2: Define the variables
Let:
- \( D_1 \) = diameter of cylinder 1
- \( D_2 \) = diameter of cylinder 2
- \( R_1 \) = radius of cylinder 1 = \( \frac{D_1}{2} \)
- \( R_2 \) = radius of cylinder 2 = \( \frac{D_2}{2} \)
- \( H_1 \) = height of cylinder 1
- \( H_2 \) = height of cylinder 2

### Step 3: Set up the ratio of diameters
From the problem, we have:
\[ \frac{D_1}{D_2} = \frac{3}{2} \]

### Step 4: Convert diameter ratio to radius ratio
Since the radius is half of the diameter, we can express the ratio of the radii as:
\[ \frac{R_1}{R_2} = \frac{D_1/2}{D_2/2} = \frac{D_1}{D_2} = \frac{3}{2} \]

### Step 5: Write the volume formula for cylinders
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi R^2 H \]
Thus, for the two cylinders, we have:
\[ V_1 = \pi R_1^2 H_1 \quad \text{and} \quad V_2 = \pi R_2^2 H_2 \]

### Step 6: Set the volumes equal
Since the volumes are equal:
\[ \pi R_1^2 H_1 = \pi R_2^2 H_2 \]
We can cancel \( \pi \) from both sides:
\[ R_1^2 H_1 = R_2^2 H_2 \]

### Step 7: Rearrange to find the height ratio
Rearranging gives us:
\[ \frac{H_1}{H_2} = \frac{R_2^2}{R_1^2} \]

### Step 8: Substitute the radius ratio
From the radius ratio \( \frac{R_1}{R_2} = \frac{3}{2} \), we can find:
\[ \frac{R_2}{R_1} = \frac{2}{3} \]

### Step 9: Square the radius ratio
Now squaring the ratio:
\[ \left(\frac{R_2}{R_1}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

### Step 10: Final height ratio
Substituting this back into our height ratio:
\[ \frac{H_1}{H_2} = \frac{4}{9} \]

### Conclusion
Thus, the ratio of the heights \( H_1 : H_2 \) is \( 4 : 9 \).