Two circular cylinders of equal volume have their heights in the ratio 1 : 2, Ratio of their radii is (Take `pi = (22)/(7)`).

Options

A. `1:4`
B. `sqrt(2):1`
C. `sqrt(4) : 1`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the radii of two circular cylinders that have equal volumes and heights in the ratio of 1:2.

### Step-by-Step Solution:

1. **Understand the Volume Formula for 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.

2. **Set Up the Volume Equations**:
Let the height of the first cylinder be \( h_1 \) and the radius be \( r_1 \). For the second cylinder, let the height be \( h_2 \) and the radius be \( r_2 \). According to the problem, the heights are in the ratio:
\[ \frac{h_1}{h_2} = \frac{1}{2} \]
This implies:
\[ h_2 = 2h_1 \]

3. **Equate the Volumes**:
Since the volumes of the two cylinders are equal, we can write:
\[ \pi r_1^2 h_1 = \pi r_2^2 h_2 \]
We can cancel \( \pi \) from both sides:
\[ r_1^2 h_1 = r_2^2 h_2 \]

4. **Substitute the Height Ratio**:
Substitute \( h_2 = 2h_1 \) into the volume equation:
\[ r_1^2 h_1 = r_2^2 (2h_1) \]
Dividing both sides by \( h_1 \) (assuming \( h_1 \neq 0 \)):
\[ r_1^2 = 2r_2^2 \]

5. **Find the Ratio of the Radii**:
Rearranging the equation gives:
\[ \frac{r_1^2}{r_2^2} = 2 \]
Taking the square root of both sides, we find:
\[ \frac{r_1}{r_2} = \sqrt{2} \]

6. **Express the Ratio**:
Thus, the ratio of the radii \( r_1 : r_2 \) is:
\[ r_1 : r_2 = \sqrt{2} : 1 \]

### Final Answer:
The ratio of the radii of the two cylinders is \( \sqrt{2} : 1 \).