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

Options

A. `1:5`
B. `4:25`
C. `16:25`
D. `25:64`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Given Ratios**:
- The volume ratio of the two cones is given as \(1:4\).
- The diameter ratio of the bases is given as \(4:5\).

2. **Finding the Radius Ratio**:
- Since the diameter ratio is \(4:5\), the radius ratio (which is half of the diameter) will also be \(4:5\).
- Therefore, if we denote the radius of the first cone as \(r_1\) and the second cone as \(r_2\), we have:
\[ \frac{r_1}{r_2} = \frac{4}{5} \]

3. **Using the Volume Formula for Cones**:
- The volume \(V\) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
- For the first cone, the volume \(V_1\) can be expressed as:
\[ V_1 = \frac{1}{3} \pi r_1^2 h_1 \]
- For the second cone, the volume \(V_2\) can be expressed as:
\[ V_2 = \frac{1}{3} \pi r_2^2 h_2 \]

4. **Setting Up the Volume Ratio**:
- According to the problem, the volume ratio is:
\[ \frac{V_1}{V_2} = \frac{1}{4} \]
- Substituting the volume expressions, we get:
\[ \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = \frac{1}{4} \]
- This simplifies to:
\[ \frac{r_1^2 h_1}{r_2^2 h_2} = \frac{1}{4} \]

5. **Substituting the Radius Ratio**:
- Now, substituting the radius ratio \( \frac{r_1}{r_2} = \frac{4}{5} \):
\[ \frac{(4)^2 h_1}{(5)^2 h_2} = \frac{1}{4} \]
- This simplifies to:
\[ \frac{16 h_1}{25 h_2} = \frac{1}{4} \]

6. **Cross Multiplying to Find the Height Ratio**:
- Cross multiplying gives:
\[ 16 h_1 = 25 h_2 \cdot \frac{1}{4} \]
- This can be rearranged to:
\[ 64 h_1 = 25 h_2 \]
- Therefore, the ratio of heights \( \frac{h_1}{h_2} \) is:
\[ \frac{h_1}{h_2} = \frac{25}{64} \]

### Final Answer:
The heights of the two cones will be in the ratio \(25:64\).