If the base radius of 2 cylinders are in the ratio `3 : 4` and their heights are in the ratio `4 : 9`, then the ratio of their volumes is:

Options

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

Answer

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

### Step 2: Set up the ratios
Let:
- The radius of the first cylinder \( r_1 \) be \( 3x \) (since the ratio of the radii is \( 3:4 \)).
- The radius of the second cylinder \( r_2 \) be \( 4x \).
- The height of the first cylinder \( h_1 \) be \( 4y \) (since the ratio of the heights is \( 4:9 \)).
- The height of the second cylinder \( h_2 \) be \( 9y \).

### Step 3: Write the volumes of both cylinders
Using the formula for volume:
- Volume of the first cylinder \( V_1 \):
\[ V_1 = \pi (r_1)^2 h_1 = \pi (3x)^2 (4y) = \pi (9x^2)(4y) = 36\pi x^2 y \]
- Volume of the second cylinder \( V_2 \):
\[ V_2 = \pi (r_2)^2 h_2 = \pi (4x)^2 (9y) = \pi (16x^2)(9y) = 144\pi x^2 y \]

### Step 4: Find the ratio of the volumes
Now, we can find the ratio of the volumes \( V_1 \) to \( V_2 \):
\[ \text{Ratio} = \frac{V_1}{V_2} = \frac{36\pi x^2 y}{144\pi x^2 y} \]
The \( \pi \), \( x^2 \), and \( y \) terms cancel out:
\[ \text{Ratio} = \frac{36}{144} = \frac{1}{4} \]

### Step 5: Conclusion
Thus, the ratio of the volumes of the two cylinders is:
\[ \text{Ratio of volumes} = 1:4 \]