The radii of two cylinders are in the ratio of 3:2 and their heights are in the ratio 3:7. The ratio of their volumes is :

Options

A. `4:7`
B. `7:4`
C. `28:27`
D. `27:28`

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: Define the Variables
Let the radius of the first cylinder be \( r_1 \) and the radius of the second cylinder be \( r_2 \). According to the problem, the ratio of the radii is given as:
\[ \frac{r_1}{r_2} = \frac{3}{2} \]
This means we can express the radii as:
\[ r_1 = 3k \quad \text{and} \quad r_2 = 2k \]
for some constant \( k \).

### Step 2: Define the Heights
Let the height of the first cylinder be \( h_1 \) and the height of the second cylinder be \( h_2 \). The ratio of the heights is given as:
\[ \frac{h_1}{h_2} = \frac{3}{7} \]
This means we can express the heights as:
\[ h_1 = 3m \quad \text{and} \quad h_2 = 7m \]
for some constant \( m \).

### Step 3: Write the Volume Formulas
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Thus, the volumes of the two cylinders can be expressed as:
\[ V_1 = \pi (r_1^2) h_1 = \pi (3k)^2 (3m) = \pi (9k^2)(3m) = 27\pi k^2 m \]
\[ V_2 = \pi (r_2^2) h_2 = \pi (2k)^2 (7m) = \pi (4k^2)(7m) = 28\pi k^2 m \]

### Step 4: Calculate the Ratio of the Volumes
Now, we can find the ratio of the volumes \( V_1 \) and \( V_2 \):
\[ \frac{V_1}{V_2} = \frac{27\pi k^2 m}{28\pi k^2 m} \]
The \( \pi k^2 m \) terms cancel out:
\[ \frac{V_1}{V_2} = \frac{27}{28} \]

### Conclusion
The ratio of the volumes of the two cylinders is:
\[ \frac{27}{28} \]