The ratio of the volume of two cones is 2:3 and the ratio of radii of their base is 1:2. The ratio of their height is

Options

A. `3:8`
B. `8:3`
C. `4:3`
D. `3:4`

Answer

To find the ratio of the heights of two cones given the ratio of their volumes and the ratio of the radii of their bases, 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.

2. **Set Up the Ratios**:
Let the volumes of the two cones be \( V_1 \) and \( V_2 \), the radii be \( r_1 \) and \( r_2 \), and the heights be \( h_1 \) and \( h_2 \).
We are given:
\[ \frac{V_1}{V_2} = \frac{2}{3} \quad \text{and} \quad \frac{r_1}{r_2} = \frac{1}{2} \]

3. **Express the Volume Ratio**:
Using the volume formula for both cones, we can express the ratio of their volumes:
\[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} \]
This simplifies to:
\[ \frac{r_1^2 h_1}{r_2^2 h_2} \]

4. **Substitute the Known Ratios**:
Substitute the known ratios into the equation:
\[ \frac{r_1^2 h_1}{r_2^2 h_2} = \frac{2}{3} \]
We know \( \frac{r_1}{r_2} = \frac{1}{2} \), so:
\[ r_1^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \quad \text{and} \quad r_2^2 = 2^2 = 4 \]

5. **Substitute \( r_1^2 \) and \( r_2^2 \)**:
Now substitute \( r_1^2 \) and \( r_2^2 \) into the volume ratio:
\[ \frac{\frac{1}{4} h_1}{4 h_2} = \frac{2}{3} \]

6. **Simplify the Equation**:
This simplifies to:
\[ \frac{h_1}{16 h_2} = \frac{2}{3} \]

7. **Cross Multiply**:
Cross multiplying gives:
\[ 3h_1 = 32h_2 \]

8. **Find the Ratio of Heights**:
Therefore, the ratio of the heights is:
\[ \frac{h_1}{h_2} = \frac{32}{3} \]

### Final Answer:
The ratio of the heights \( h_1 : h_2 \) is \( 32 : 3 \).