The base radi of two right circular cylinders are in the ratio of 1:2 and their respective heights are in the ratio of 4:1 The volume of the cylinders are in the ratio of:

Options

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

Answer

To find the ratio of the volumes of two right circular 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 right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height of the cylinder.

### Step 2: Assign variables based on the given ratios
Let the radius of the first cylinder be \( r_1 = 1x \) and the radius of the second cylinder be \( r_2 = 2x \) (since the ratio of the radii is 1:2).

Let the height of the first cylinder be \( h_1 = 4y \) and the height of the second cylinder be \( h_2 = 1y \) (since the ratio of the heights is 4:1).

### Step 3: Calculate the volumes of both cylinders
Using the volume formula:

For the first cylinder:
\[ V_1 = \pi (r_1)^2 h_1 = \pi (1x)^2 (4y) = \pi (1^2) x^2 (4y) = 4\pi x^2 y \]

For the second cylinder:
\[ V_2 = \pi (r_2)^2 h_2 = \pi (2x)^2 (1y) = \pi (2^2) x^2 (1y) = 4\pi x^2 y \]

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

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