The volume of two cylinders are as a : b and their heights are c : d Find the ratio of their diameters ?

Options

A. `(ab)/(bc)`
B. `(ad^(2))/(ac^(2))`
C. `sqrt((ad)/(bc))`
D. `sqrt((a)/(b))xx(c )/(d)`

Answer

To find the ratio of the diameters of two cylinders given the ratio of their volumes and heights, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
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. **Express Radius in Terms of Diameter**:
The radius \( r \) can be expressed in terms of the diameter \( d \):
\[ r = \frac{d}{2} \]
Therefore, the volume can also be expressed as:
\[ V = \pi \left(\frac{d}{2}\right)^2 h = \frac{\pi d^2 h}{4} \]

3. **Set Up the Volumes for Both Cylinders**:
Let the volumes of the first and second cylinders be \( V_1 \) and \( V_2 \):
\[ V_1 = \frac{\pi d_1^2 h_1}{4} \]
\[ V_2 = \frac{\pi d_2^2 h_2}{4} \]

4. **Use the Given Ratios**:
We are given that the ratio of the volumes is:
\[ \frac{V_1}{V_2} = \frac{A}{B} \]
and the ratio of the heights is:
\[ \frac{h_1}{h_2} = \frac{C}{D} \]

5. **Substitute the Volumes into the Ratio**:
Substitute the expressions for \( V_1 \) and \( V_2 \) into the volume ratio:
\[ \frac{\frac{\pi d_1^2 h_1}{4}}{\frac{\pi d_2^2 h_2}{4}} = \frac{A}{B} \]
The \( \frac{\pi}{4} \) cancels out:
\[ \frac{d_1^2 h_1}{d_2^2 h_2} = \frac{A}{B} \]

6. **Substitute the Height Ratio**:
Now substitute \( h_1 = \frac{C}{D} h_2 \) into the equation:
\[ \frac{d_1^2 \left(\frac{C}{D} h_2\right)}{d_2^2 h_2} = \frac{A}{B} \]
The \( h_2 \) cancels out:
\[ \frac{d_1^2 \frac{C}{D}}{d_2^2} = \frac{A}{B} \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{d_1^2}{d_2^2} = \frac{A D}{B C} \]

8. **Taking the Square Root**:
Taking the square root of both sides, we find the ratio of the diameters:
\[ \frac{d_1}{d_2} = \sqrt{\frac{A D}{B C}} \]

### Final Answer:
The ratio of the diameters \( d_1 : d_2 \) is:
\[ \frac{d_1}{d_2} = \sqrt{\frac{A D}{B C}} \]