The respective ratio of the heights of the right circular cylinders A and B is `1:2`. If the respective ratio of the volumes of the cylinders A and B is `2 : 9`, what is the respective ratio of radii of cylinders A and B?

Options

A. `3:4`
B. `1:3`
C. `2:3`
D. `2:5`

Answer

To solve the problem, we need to find the ratio of the radii of two right circular cylinders A and B, given the ratios of their heights and volumes.

### Step-by-Step Solution:

1. **Define Variables:**
Let the height of cylinder A be \( h_A \) and the height of cylinder B be \( h_B \). According to the problem, the ratio of their heights is given as:
\[ h_A : h_B = 1 : 2 \]
We can express this as:
\[ h_A = x \quad \text{and} \quad h_B = 2x \]

2. **Volume Formula:**
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Let the radius of cylinder A be \( r_A \) and the radius of cylinder B be \( r_B \). Therefore, the volumes can be expressed as:
\[ V_A = \pi r_A^2 h_A = \pi r_A^2 x \]
\[ V_B = \pi r_B^2 h_B = \pi r_B^2 (2x) \]

3. **Volume Ratio:**
The problem states that the ratio of the volumes of cylinders A and B is:
\[ V_A : V_B = 2 : 9 \]
This can be written as:
\[ \frac{V_A}{V_B} = \frac{2}{9} \]
Substituting the expressions for \( V_A \) and \( V_B \):
\[ \frac{\pi r_A^2 x}{\pi r_B^2 (2x)} = \frac{2}{9} \]
The \( \pi \) and \( x \) cancel out:
\[ \frac{r_A^2}{2r_B^2} = \frac{2}{9} \]

4. **Cross Multiplying:**
Cross-multiplying gives us:
\[ 9r_A^2 = 4r_B^2 \]

5. **Finding the Ratio of Radii:**
Rearranging the equation gives:
\[ \frac{r_A^2}{r_B^2} = \frac{4}{9} \]
Taking the square root of both sides:
\[ \frac{r_A}{r_B} = \frac{2}{3} \]

6. **Final Ratio:**
Thus, the respective ratio of the radii of cylinders A and B is:
\[ r_A : r_B = 2 : 3 \]

### Conclusion:
The respective ratio of the radii of cylinders A and B is \( 2 : 3 \).