If the base radii of two cylinders are in the ratio 3:4 and their heights are in the ratio 4: 9. then the ratio of their respective volumes is :

Options

A. `2:1`
B. `1:4`
C. `4:1`
D. `1:2`

Answer

To find the ratio of the volumes of two cylinders given the ratios of their base 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 base radius and \( h \) is the height of the cylinder.

### Step 2: Set up the ratios
We are given:
- The ratio of the base radii of the two cylinders \( r_1 : r_2 = 3 : 4 \)
- The ratio of the heights of the two cylinders \( h_1 : h_2 = 4 : 9 \)

### Step 3: Express the radii and heights in terms of variables
Let:
- \( r_1 = 3k \) and \( r_2 = 4k \) for some constant \( k \)
- \( h_1 = 4m \) and \( h_2 = 9m \) for some constant \( m \)

### Step 4: Write the volumes of the cylinders
Using the formula for volume, we can express the volumes of the two cylinders:
- Volume of the first cylinder \( V_1 = \pi (r_1^2) h_1 = \pi (3k)^2 (4m) = \pi (9k^2)(4m) = 36\pi k^2 m \)
- Volume of the second cylinder \( V_2 = \pi (r_2^2) h_2 = \pi (4k)^2 (9m) = \pi (16k^2)(9m) = 144\pi k^2 m \)

### Step 5: Find the ratio of the volumes
Now we can find the ratio of the volumes \( V_1 : V_2 \):
\[ \frac{V_1}{V_2} = \frac{36\pi k^2 m}{144\pi k^2 m} \]
The \( \pi k^2 m \) terms cancel out:
\[ \frac{V_1}{V_2} = \frac{36}{144} = \frac{1}{4} \]

### Step 6: State the final ratio
Thus, the ratio of the volumes of the two cylinders is:
\[ V_1 : V_2 = 1 : 4 \]

### Conclusion
The final answer is that the ratio of the volumes of the two cylinders is \( 1 : 4 \).