The ratio of the radii of two cylinders is `2 : 3`, and the ratio of their heights is `5 : 3`. The ratio of their volumes will be

Options

A. `9:4`
B. `20:27`
C. `4:9`
D. `27:20`

Answer

To find the ratio of the volumes of two cylinders given the ratios of their radii and heights, we can follow these steps:

### Step 1: Understand the formula for the volume of a cylinder.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Identify the given ratios.
We are given:
- The ratio of the radii of the two cylinders \( r_1 : r_2 = 2 : 3 \)
- The ratio of the heights of the two cylinders \( h_1 : h_2 = 5 : 3 \)

### Step 3: Express the radii and heights in terms of variables.
Let:
- \( r_1 = 2k \) and \( r_2 = 3k \) for some constant \( k \)
- \( h_1 = 5m \) and \( h_2 = 3m \) for some constant \( m \)

### Step 4: Substitute the values into the volume formula.
Now, we can express the volumes of the two cylinders:
\[ V_1 = \pi (r_1)^2 h_1 = \pi (2k)^2 (5m) = \pi (4k^2)(5m) = 20\pi k^2 m \]
\[ V_2 = \pi (r_2)^2 h_2 = \pi (3k)^2 (3m) = \pi (9k^2)(3m) = 27\pi k^2 m \]

### Step 5: Find the ratio of the volumes.
Now we can find the ratio of the volumes \( V_1 : V_2 \):
\[ \frac{V_1}{V_2} = \frac{20\pi k^2 m}{27\pi k^2 m} \]
The \( \pi \), \( k^2 \), and \( m \) cancel out:
\[ \frac{V_1}{V_2} = \frac{20}{27} \]

### Conclusion
Thus, the ratio of the volumes of the two cylinders is:
\[ \boxed{20 : 27} \]

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