The radii of two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3 . The ratio of their volumes is

Options

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

Explanation

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: Define the radii and heights
Let the radius of the first cylinder be \( r_1 = 2x \) and the radius of the second cylinder be \( r_2 = 3x \). Here, \( x \) is a common multiplier.

Let the height of the first cylinder be \( h_1 = 5y \) and the height of the second cylinder be \( h_2 = 3y \). Here, \( y \) is another common multiplier.

### Step 2: Write the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]

### Step 3: Calculate the volumes of both cylinders
For the first cylinder:
\[ V_1 = \pi (r_1)^2 h_1 = \pi (2x)^2 (5y) = \pi (4x^2)(5y) = 20\pi x^2 y \]

For the second cylinder:
\[ V_2 = \pi (r_2)^2 h_2 = \pi (3x)^2 (3y) = \pi (9x^2)(3y) = 27\pi x^2 y \]

### Step 4: Find the ratio of the volumes
Now, we can find the ratio of the volumes \( V_1 \) and \( V_2 \):
\[ \text{Ratio of volumes} = \frac{V_1}{V_2} = \frac{20\pi x^2 y}{27\pi x^2 y} \]

### Step 5: Simplify the ratio
The \( \pi \), \( x^2 \), and \( y \) terms cancel out:
\[ \frac{V_1}{V_2} = \frac{20}{27} \]

### Final Answer
Thus, the ratio of the volumes of the two cylinders is \( \frac{20}{27} \).

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