If the ratio of volumes of two cones is 2:3 and the ratio of the radii of their bases is 1:2, then the ratio of their heights will be.
यदि दो शंकुओं के आयतनों का अनुपात 2:3 है और उनके आधारों की त्रिज्याओं का अनुपात 1 : 2 है, तो उनकी ऊँचाईयों का अनुपात होगा

Options

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

Explanation

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 of a Cone**:
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. **Set Up the Ratios**:
We are given the ratio of the volumes of two cones \( V_1 : V_2 = 2 : 3 \) and the ratio of the radii of their bases \( r_1 : r_2 = 1 : 2 \).

3. **Express the Volumes in Terms of the Radii and Heights**:
Using the volume formula for both cones, we can write:
\[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} \]
This simplifies to:
\[ \frac{r_1^2 h_1}{r_2^2 h_2} \]

4. **Substitute the Known Ratios**:
We know \( \frac{V_1}{V_2} = \frac{2}{3} \) and \( \frac{r_1}{r_2} = \frac{1}{2} \). Therefore, we can express \( r_1^2 \) and \( r_2^2 \):
\[ r_1^2 = \left(\frac{1}{2} r_2\right)^2 = \frac{1}{4} r_2^2 \]
Now substituting this into the volume ratio gives:
\[ \frac{\frac{1}{4} r_2^2 h_1}{r_2^2 h_2} = \frac{2}{3} \]

5. **Simplify the Equation**:
The \( r_2^2 \) cancels out:
\[ \frac{1}{4} \frac{h_1}{h_2} = \frac{2}{3} \]

6. **Cross Multiply to Solve for the Height Ratio**:
Cross multiplying gives:
\[ 3h_1 = 8h_2 \]
Therefore, we can express the ratio of the heights:
\[ \frac{h_1}{h_2} = \frac{8}{3} \]

7. **Conclusion**:
The ratio of the heights of the two cones is:
\[ h_1 : h_2 = 8 : 3 \]