The heights of two cylinders are in the ratio of 3:1 . If the volumes of two cylinders be same , the ratio of their respective radii are :

Options

A. `sqrt(3) :1`
B. `1: sqrt(3) `
C. `1:9`
D. none of these

Answer

To solve the problem, we need to find the ratio of the respective radii of two cylinders given that their heights are in the ratio of 3:1 and their volumes are the same.

### Step-by-Step Solution:

1. **Define the Heights**:
Let the height of the first cylinder be \( h_1 \) and the height of the second cylinder be \( h_2 \). According to the problem, the ratio of their heights is given as:
\[ h_1 : h_2 = 3 : 1 \]
This can be expressed as:
\[ \frac{h_1}{h_2} = \frac{3}{1} \]

2. **Volume of the Cylinders**:
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Therefore, the volumes of the two cylinders can be expressed as:
- Volume of Cylinder 1: \( V_1 = \pi r_1^2 h_1 \)
- Volume of Cylinder 2: \( V_2 = \pi r_2^2 h_2 \)

3. **Set the Volumes Equal**:
Since the volumes of the two cylinders are the same, we can set their volume equations equal to each other:
\[ \pi r_1^2 h_1 = \pi r_2^2 h_2 \]
We can cancel \( \pi \) from both sides:
\[ r_1^2 h_1 = r_2^2 h_2 \]

4. **Substitute the Height Ratio**:
From the height ratio \( \frac{h_1}{h_2} = \frac{3}{1} \), we can express \( h_1 \) in terms of \( h_2 \):
\[ h_1 = 3h_2 \]
Substitute \( h_1 \) into the volume equation:
\[ r_1^2 (3h_2) = r_2^2 h_2 \]

5. **Simplify the Equation**:
We can divide both sides by \( h_2 \) (assuming \( h_2 \neq 0 \)):
\[ 3r_1^2 = r_2^2 \]

6. **Express the Radii Ratio**:
Rearranging gives us:
\[ \frac{r_1^2}{r_2^2} = \frac{1}{3} \]
Taking the square root of both sides:
\[ \frac{r_1}{r_2} = \frac{1}{\sqrt{3}} \]

7. **Final Ratio**:
Thus, the ratio of the respective radii \( r_1 : r_2 \) is:
\[ r_1 : r_2 = 1 : \sqrt{3} \]

### Conclusion:
The ratio of the respective radii of the two cylinders is \( 1 : \sqrt{3} \).

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