Assertion (A) : The radii of two cone are in the ratio 2:3 and their volumes in the ratio 1:3. Then ratio of their heights is 3:2.
Reason (R) : Volume of cone `= (1)/(3) pi r^(2)h`

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is not the correct explanation of A.
C. A is true but R is false
D. A is false but R is true

Answer

To solve the problem step by step, we will analyze the assertion and reason provided, and then derive the necessary relationships using the formula for the volume of a cone.

### Step-by-Step Solution:

1. **Identify the Given Ratios:**
- The ratio of the radii of two cones is given as \( R_1 : R_2 = 2 : 3 \).
- The ratio of their volumes is given as \( V_1 : V_2 = 1 : 3 \).

2. **Use the Volume Formula for a Cone:**
- The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
- For cone 1 (with radius \( R_1 \) and height \( H_1 \)):
\[ V_1 = \frac{1}{3} \pi R_1^2 H_1 \]
- For cone 2 (with radius \( R_2 \) and height \( H_2 \)):
\[ V_2 = \frac{1}{3} \pi R_2^2 H_2 \]

3. **Set Up the Volume Ratio:**
- From the volume formula, we can write the ratio of the volumes:
\[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi R_1^2 H_1}{\frac{1}{3} \pi R_2^2 H_2} = \frac{R_1^2 H_1}{R_2^2 H_2} \]
- Given \( \frac{V_1}{V_2} = \frac{1}{3} \), we can write:
\[ \frac{R_1^2 H_1}{R_2^2 H_2} = \frac{1}{3} \]

4. **Substitute the Ratios of Radii:**
- From the given ratio of radii \( \frac{R_1}{R_2} = \frac{2}{3} \), we can express \( R_1^2 \) and \( R_2^2 \):
\[ \frac{R_1^2}{R_2^2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

5. **Substitute into the Volume Ratio:**
- Substitute \( \frac{R_1^2}{R_2^2} \) into the volume ratio equation:
\[ \frac{4}{9} \cdot \frac{H_1}{H_2} = \frac{1}{3} \]

6. **Solve for the Ratio of Heights:**
- Rearranging gives:
\[ \frac{H_1}{H_2} = \frac{1}{3} \cdot \frac{9}{4} = \frac{3}{4} \]
- Thus, the ratio of the heights \( H_1 : H_2 = 3 : 4 \).

7. **Conclusion:**
- The assertion states that the ratio of heights is \( 3 : 2 \), which is incorrect. The correct ratio is \( 3 : 4 \).
- The reason provided is correct as it states the formula for the volume of a cone.

### Final Answer:
- The assertion (A) is **false** and the reason (R) is **true**.