The volume of two cylinders are equal and their heights are in the ratio 1:3. Then, the ratio of their radius are

Options

A. `4:sqrt3`
B. `3:2sqrt3`
C. `2:sqrt3`
D. `3:sqrt3`

Answer

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

### Step-by-Step Solution:

1. **Understand 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.

2. **Set Up the Problem**:
Let the height of the first cylinder be \( h \) and the height of the second cylinder be \( 3h \) (since the heights are in the ratio 1:3). Let the radius of the first cylinder be \( r \) and the radius of the second cylinder be \( r' \).

3. **Write the Volume Equations**:
The volume of the first cylinder can be expressed as:
\[ V_1 = \pi r^2 h \]
The volume of the second cylinder can be expressed as:
\[ V_2 = \pi (r')^2 (3h) \]

4. **Set the Volumes Equal**:
Since the volumes are equal, we can set the two equations equal to each other:
\[ \pi r^2 h = \pi (r')^2 (3h) \]

5. **Cancel Common Terms**:
We can cancel \( \pi \) and \( h \) from both sides (assuming \( h \neq 0 \)):
\[ r^2 = 3 (r')^2 \]

6. **Rearrange the Equation**:
To find the ratio of the radii, we can rearrange this equation:
\[ \frac{r^2}{(r')^2} = 3 \]

7. **Take the Square Root**:
Taking the square root of both sides gives:
\[ \frac{r}{r'} = \sqrt{3} \]

8. **Express the Ratio**:
Thus, the ratio of the radii \( r \) and \( r' \) is:
\[ r : r' = \sqrt{3} : 1 \]

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