The ration of the volume of the two cylinder, if the height of the first is equal to the diameter of the secondand the diameterof the first is equal to the height of the seond is

Options

A. 2r/h
B. 6r/h
C. 10r/h
D. r/h

Answer

To solve the problem of finding the ratio of the volumes of two cylinders given the specific relationships between their dimensions, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the height of the first cylinder be \( h \) and the radius be \( r \).
- Let the height of the second cylinder be \( H \) and the radius be \( R \).

2. **Establish Relationships:**
- According to the problem, the height of the first cylinder is equal to the diameter of the second cylinder. The diameter of the second cylinder is \( 2R \). Therefore, we can write:
\[ h = 2R \]
- The diameter of the first cylinder is equal to the height of the second cylinder. The diameter of the first cylinder is \( 2r \). Therefore, we can write:
\[ 2r = H \]

3. **Express \( H \) in terms of \( r \):**
- From the equation \( 2r = H \), we can express \( H \) as:
\[ H = 2r \]

4. **Substitute \( H \) into the first equation:**
- Now, substituting \( H \) into the first relationship \( h = 2R \):
\[ h = 2R \]

5. **Calculate the Volumes of the Cylinders:**
- The volume \( V_1 \) of the first cylinder is given by:
\[ V_1 = \pi r^2 h \]
- The volume \( V_2 \) of the second cylinder is given by:
\[ V_2 = \pi R^2 H \]

6. **Substituting \( h \) and \( H \):**
- Substitute \( h = 2R \) and \( H = 2r \) into the volume formulas:
\[ V_1 = \pi r^2 (2R) = 2\pi r^2 R \]
\[ V_2 = \pi R^2 (2r) = 2\pi R^2 r \]

7. **Find the Ratio of Volumes:**
- Now, we can find the ratio of the volumes \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{2\pi r^2 R}{2\pi R^2 r} \]
- Simplifying this gives:
\[ \frac{V_1}{V_2} = \frac{r^2 R}{R^2 r} = \frac{r}{R} \]

8. **Final Ratio:**
- Therefore, the ratio of the volumes of the two cylinders is:
\[ \frac{V_1}{V_2} = \frac{r}{R} \]