In `Delta ABC`, circumrdius is 3 inradius is 1.5 units. The value of a `acot^(2)A+b^(2)cot^(3)B+c^(3)cot^(4)C` is

Options

A. `13 sqrt(3)`
B. `11sqrt(6)`
C. 21
D. none of these

Explanation

To solve the problem, we need to find the value of \( a \cot^2 A + b^2 \cot^3 B + c^3 \cot^4 C \) in triangle \( ABC \) where the circumradius \( R = 3 \) and the inradius \( r = 1.5 \).

### Step-by-Step Solution:

1. **Identify the relationship between circumradius and inradius**:
Given that \( R = 3 \) and \( r = 1.5 \), we can check the relationship:
\[ \frac{R}{r} = \frac{3}{1.5} = 2 \]
This indicates that triangle \( ABC \) is equilateral since for an equilateral triangle, \( R = 2r \).

**Hint**: Check the ratio of circumradius to inradius to determine the type of triangle.

2. **Determine the angles of the triangle**:
Since triangle \( ABC \) is equilateral, all angles \( A, B, C \) are equal:
\[ A = B = C = 60^\circ \]

**Hint**: In an equilateral triangle, all angles are equal.

3. **Calculate the sides of the triangle**:
The sides \( a, b, c \) can be calculated using the formula:
\[ a = 2R \sin A \]
Since \( A = 60^\circ \):
\[ a = 2 \times 3 \times \sin(60^\circ) = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3} \]
Similarly, \( b = c = 3\sqrt{3} \).

**Hint**: Use the sine rule to find the sides based on the circumradius.

4. **Calculate \( \cot A, \cot B, \cot C \)**:
Since \( A = B = C = 60^\circ \):
\[ \cot A = \cot 60^\circ = \frac{1}{\sqrt{3}} \]

**Hint**: Remember that \( \cot(60^\circ) = \frac{1}{\tan(60^\circ)} \).

5. **Substitute values into the expression**:
Now we substitute \( a, b, c \) and \( \cot A, \cot B, \cot C \) into the expression:
\[ a \cot^2 A + b^2 \cot^3 B + c^3 \cot^4 C \]
This becomes:
\[ (3\sqrt{3}) \left(\frac{1}{\sqrt{3}}\right)^2 + (3\sqrt{3})^2 \left(\frac{1}{\sqrt{3}}\right)^3 + (3\sqrt{3})^3 \left(\frac{1}{\sqrt{3}}\right)^4 \]

6. **Calculate each term**:
- First term:
\[ (3\sqrt{3}) \cdot \frac{1}{3} = \sqrt{3} \]
- Second term:
\[ (3\sqrt{3})^2 \cdot \frac{1}{3\sqrt{3}} = 27 \cdot \frac{1}{3\sqrt{3}} = 9\sqrt{3} \]
- Third term:
\[ (3\sqrt{3})^3 \cdot \frac{1}{9} = 27\sqrt{3} \cdot \frac{1}{9} = 3\sqrt{3} \]

7. **Combine the results**:
Now, combine all the terms:
\[ \sqrt{3} + 9\sqrt{3} + 3\sqrt{3} = 13\sqrt{3} \]

### Final Answer:
The value of \( a \cot^2 A + b^2 \cot^3 B + c^3 \cot^4 C \) is \( 13\sqrt{3} \).