The ratio of the volumes of two right circular cylinders A and B `(x)/(y)` is and the ratio of their heights is a : b. What is the ratio of the radii of A and B?

Options

A. `(yb)/(xa)`
B. `sqrt((xa)/(yb))`
C. `(xb)/(ya)`
D. `sqrt((xb)/(ya))`

Answer

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

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
The volume \( V \) of a right circular 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 Ratios**:
Let the volume of cylinder A be \( V_A \) and the volume of cylinder B be \( V_B \). According to the problem:
\[ \frac{V_A}{V_B} = \frac{x}{y} \]
Let the heights of cylinders A and B be \( h_A \) and \( h_B \) respectively. The problem states that:
\[ \frac{h_A}{h_B} = \frac{a}{b} \]

3. **Express Volumes in Terms of Radii and Heights**:
Using the volume formula, we can express the volumes of the cylinders as:
\[ V_A = \pi r_A^2 h_A \quad \text{and} \quad V_B = \pi r_B^2 h_B \]

4. **Substitute the Volume Ratios**:
Now, substituting the expressions for volumes into the ratio:
\[ \frac{V_A}{V_B} = \frac{\pi r_A^2 h_A}{\pi r_B^2 h_B} = \frac{r_A^2 h_A}{r_B^2 h_B} \]
This simplifies to:
\[ \frac{r_A^2}{r_B^2} \cdot \frac{h_A}{h_B} = \frac{x}{y} \]

5. **Substitute the Height Ratio**:
We know that:
\[ \frac{h_A}{h_B} = \frac{a}{b} \]
Substituting this into the equation gives:
\[ \frac{r_A^2}{r_B^2} \cdot \frac{a}{b} = \frac{x}{y} \]

6. **Rearranging the Equation**:
Rearranging the equation to isolate the ratio of the squares of the radii:
\[ \frac{r_A^2}{r_B^2} = \frac{x}{y} \cdot \frac{b}{a} \]

7. **Taking the Square Root**:
To find the ratio of the radii \( \frac{r_A}{r_B} \), we take the square root of both sides:
\[ \frac{r_A}{r_B} = \sqrt{\frac{x}{y} \cdot \frac{b}{a}} = \frac{r_A}{r_B} = \frac{\sqrt{bx}}{\sqrt{ay}} \]

8. **Final Ratio of Radii**:
Thus, the ratio of the radii of cylinders A and B is:
\[ \frac{r_A}{r_B} = \frac{b\sqrt{x}}{a\sqrt{y}} \]

### Conclusion:
The correct answer is:
\[ \frac{r_A}{r_B} = \frac{b\sqrt{x}}{a\sqrt{y}} \quad \text{(Option 4: } \frac{xb}{ya} \text{)} \]