If the volumes of two right circular cones are in the ratio 4 : 1 and their diameters are in the ratio 5 : 4, then the ratio of their heights is

Options

A. `25 : 16`
B. `25 : 64`
C. `64 : 25`
D. `16 : 25`

Answer

To solve the problem, we need to find the ratio of the heights of two right circular cones given the ratios of their volumes and diameters.

### Step-by-Step Solution:

1. **Understanding the Volume of a Cone**:
The formula for the volume \( V \) of a right circular cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

2. **Given Ratios**:
- The volumes of the two cones are in the ratio \( 4:1 \).
- The diameters of the two cones are in the ratio \( 5:4 \).

3. **Finding the Radius Ratio**:
Since the diameter ratio is \( 5:4 \), the radius ratio \( r_1:r_2 \) will also be:
\[ r_1:r_2 = \frac{5}{2} : \frac{4}{2} = 5:4 \]

4. **Setting Up the Volume Ratio**:
Let the volumes of the two cones be \( V_1 \) and \( V_2 \). According to the problem:
\[ \frac{V_1}{V_2} = \frac{4}{1} \]
Substituting the volume formula:
\[ \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = \frac{4}{1} \]
The \( \frac{1}{3} \pi \) cancels out:
\[ \frac{r_1^2 h_1}{r_2^2 h_2} = 4 \]

5. **Substituting the Radius Ratio**:
Using the radius ratio \( r_1:r_2 = 5:4 \):
\[ r_1 = 5k \quad \text{and} \quad r_2 = 4k \quad \text{for some } k \]
Therefore:
\[ r_1^2 = (5k)^2 = 25k^2 \quad \text{and} \quad r_2^2 = (4k)^2 = 16k^2 \]

6. **Substituting into the Volume Ratio**:
Now substituting \( r_1^2 \) and \( r_2^2 \) into the volume ratio:
\[ \frac{25k^2 h_1}{16k^2 h_2} = 4 \]
The \( k^2 \) cancels out:
\[ \frac{25 h_1}{16 h_2} = 4 \]

7. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 25 h_1 = 64 h_2 \]

8. **Finding the Height Ratio**:
Rearranging gives:
\[ \frac{h_1}{h_2} = \frac{64}{25} \]

### Final Answer:
The ratio of the heights \( h_1:h_2 \) is \( 64:25 \).