Ratio between heights of two cylinder in the ratio 3:5. Their volumes are in the ratio 27:80. Find the ratio between their radius.

Options

A. a. 2:3
B. b. 3:2
C. c. 3:4
D. d. 4:3

Answer

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

### Step-by-Step Solution:

1. **Understand the Given Ratios:**
- The ratio of the heights of the two cylinders is given as \( h_1 : h_2 = 3 : 5 \).
- The ratio of the volumes of the two cylinders is given as \( V_1 : V_2 = 27 : 80 \).

2. **Volume of a Cylinder Formula:**
- The volume \( V \) of a cylinder is calculated using the formula:
\[ V = \pi r^2 h \]
- For Cylinder 1, let the radius be \( r_1 \) and height be \( h_1 \).
- For Cylinder 2, let the radius be \( r_2 \) and height be \( h_2 \).

3. **Set Up the Volume Ratio:**
- Using the volume formula, we can express the volumes of the two cylinders:
\[ V_1 = \pi r_1^2 h_1 \quad \text{and} \quad V_2 = \pi r_2^2 h_2 \]
- The ratio of the volumes can be written as:
\[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} \]
- This simplifies to:
\[ \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \]

4. **Substitute the Given Ratios:**
- We know \( \frac{V_1}{V_2} = \frac{27}{80} \) and the height ratio \( \frac{h_1}{h_2} = \frac{3}{5} \).
- Substitute \( h_1 = 3k \) and \( h_2 = 5k \) for some \( k \):
\[ \frac{27}{80} = \frac{r_1^2 \cdot 3k}{r_2^2 \cdot 5k} \]
- The \( k \) cancels out:
\[ \frac{27}{80} = \frac{3r_1^2}{5r_2^2} \]

5. **Cross Multiply to Solve for Radii:**
- Cross multiplying gives:
\[ 27 \cdot 5 r_2^2 = 80 \cdot 3 r_1^2 \]
- Simplifying this:
\[ 135 r_2^2 = 240 r_1^2 \]

6. **Rearranging for the Ratio of Radii:**
- Dividing both sides by \( r_1^2 r_2^2 \):
\[ \frac{r_1^2}{r_2^2} = \frac{135}{240} \]
- Simplifying \( \frac{135}{240} \):
\[ \frac{135 \div 15}{240 \div 15} = \frac{9}{16} \]

7. **Taking the Square Root:**
- To find the ratio of the radii, take the square root:
\[ \frac{r_1}{r_2} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

8. **Final Ratio:**
- Thus, the ratio of the radii of the two cylinders is:
\[ r_1 : r_2 = 3 : 4 \]

### Conclusion:
The ratio between the radii of the two cylinders is \( 3 : 4 \).