The ratio of the volumes of two cones of same base is 8 : 27. Find the ratio of their heights.

Answer

To solve the problem of finding the ratio of the heights of two cones given the ratio of their volumes, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify Given Information**:
We are given that the ratio of the volumes of two cones is:
\[ V_1 : V_2 = 8 : 27 \]

3. **Set Up the Volume Ratio**:
Let the volumes of the two cones be \( V_1 \) and \( V_2 \). According to the volume formula, we can express the volumes as:
\[ V_1 = \frac{1}{3} \pi r_1^2 h_1 \quad \text{and} \quad V_2 = \frac{1}{3} \pi r_2^2 h_2 \]
Since the cones have the same base, we have \( r_1 = r_2 \).

4. **Simplify the Volume Ratio**:
Since \( r_1 = r_2 \), we can denote the common radius as \( r \). Thus, we can rewrite the volume ratio as:
\[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r^2 h_1}{\frac{1}{3} \pi r^2 h_2} \]
The \( \frac{1}{3} \pi r^2 \) terms cancel out, leading to:
\[ \frac{h_1}{h_2} = \frac{V_1}{V_2} \]

5. **Substitute the Given Volume Ratio**:
We know that \( \frac{V_1}{V_2} = \frac{8}{27} \). Therefore, we have:
\[ \frac{h_1}{h_2} = \frac{8}{27} \]

6. **Conclusion**:
The ratio of the heights of the two cones is:
\[ h_1 : h_2 = 8 : 27 \]

### Final Answer:
The ratio of the heights of the two cones is \( 8 : 27 \).