If the volumes of two right circular cones are in the ratio 1 : 3 and their diameters are in the ratio 3:5, then the ratio of their heights is

Options

A. `25:27`
B. `1:5`
C. `3:5`
D. `5:27`

Answer

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

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- The volumes of the two cones are in the ratio \( V_1 : V_2 = 1 : 3 \).
- The diameters of the two cones are in the ratio \( D_1 : D_2 = 3 : 5 \).

2. **Relate Diameter to Radius**:
- The radius is half of the diameter. Therefore, the ratio of the radii \( r_1 : r_2 \) will be:
\[ r_1 : r_2 = \frac{D_1}{2} : \frac{D_2}{2} = 3 : 5 \]
- This means \( r_1 : r_2 = 3 : 5 \).

3. **Use the Volume Formula for Cones**:
- The volume \( V \) of a right circular cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
- For the two cones, we can write:
\[ V_1 = \frac{1}{3} \pi r_1^2 h_1 \quad \text{and} \quad V_2 = \frac{1}{3} \pi r_2^2 h_2 \]

4. **Set Up the Volume Ratio**:
- From the volume ratio \( V_1 : V_2 = 1 : 3 \), we have:
\[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = \frac{r_1^2 h_1}{r_2^2 h_2} = \frac{1}{3} \]

5. **Substitute the Radius Ratio**:
- Substitute \( r_1 : r_2 = 3 : 5 \) into the equation:
\[ \frac{(3)^2 h_1}{(5)^2 h_2} = \frac{1}{3} \]
- This simplifies to:
\[ \frac{9 h_1}{25 h_2} = \frac{1}{3} \]

6. **Cross-Multiply to Solve for the Height Ratio**:
- Cross-multiplying gives:
\[ 9 h_1 = \frac{25}{3} h_2 \]
- Rearranging gives:
\[ \frac{h_1}{h_2} = \frac{25}{27} \]

7. **Final Ratio of Heights**:
- Thus, the ratio of the heights \( h_1 : h_2 \) is:
\[ h_1 : h_2 = 25 : 27 \]

### Final Answer:
The ratio of the heights of the two cones is \( 25 : 27 \).