In `Delta ABC`, If `angle C = 3 angle A, BC = 27, and AB =48`. Then the value of AC is ______

Explanation

To solve the problem, we will use the Sine Rule in triangle \( ABC \). Given the information:

- \( \angle C = 3 \angle A \)
- \( BC = 27 \) (let's denote this as side \( a \))
- \( AB = 48 \) (let's denote this as side \( c \))
- We need to find \( AC \) (let's denote this as side \( b \))

### Step 1: Set Up the Sine Rule
According to the Sine Rule, we have:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

### Step 2: Express \( \sin C \) in terms of \( \sin A \)
Since \( \angle C = 3 \angle A \), we can express \( \sin C \) using the sine triple angle formula:

\[ \sin C = \sin(3A) = 3 \sin A - 4 \sin^3 A \]

### Step 3: Substitute Known Values in Sine Rule
Using the Sine Rule, we can write:

\[ \frac{27}{\sin A} = \frac{48}{\sin C} \]

Substituting \( \sin C \):

\[ \frac{27}{\sin A} = \frac{48}{3 \sin A - 4 \sin^3 A} \]

### Step 4: Cross Multiply
Cross multiplying gives us:

\[ 27(3 \sin A - 4 \sin^3 A) = 48 \sin A \]

### Step 5: Rearrange the Equation
Rearranging the equation:

\[ 81 \sin A - 108 \sin^3 A = 48 \sin A \]

This simplifies to:

\[ 33 \sin A - 108 \sin^3 A = 0 \]

### Step 6: Factor Out \( \sin A \)
Factoring out \( \sin A \):

\[ \sin A (33 - 108 \sin^2 A) = 0 \]

### Step 7: Solve for \( \sin A \)
Since \( \sin A \neq 0 \) in a triangle, we have:

\[ 33 - 108 \sin^2 A = 0 \]

Solving for \( \sin^2 A \):

\[ 108 \sin^2 A = 33 \implies \sin^2 A = \frac{33}{108} = \frac{11}{36} \]

Taking the square root:

\[ \sin A = \frac{\sqrt{11}}{6} \]

### Step 8: Find \( \sin C \)
Now we can find \( \sin C \):

\[ \sin C = 3 \sin A - 4 \sin^3 A \]

Calculating \( \sin^3 A \):

\[ \sin^3 A = \left(\frac{\sqrt{11}}{6}\right)^3 = \frac{11\sqrt{11}}{216} \]

Substituting back:

\[ \sin C = 3 \cdot \frac{\sqrt{11}}{6} - 4 \cdot \frac{11\sqrt{11}}{216} \]

Calculating:

\[ \sin C = \frac{3\sqrt{11}}{6} - \frac{44\sqrt{11}}{216} = \frac{3\sqrt{11}}{6} - \frac{11\sqrt{11}}{54} = \frac{27\sqrt{11}}{54} - \frac{11\sqrt{11}}{54} = \frac{16\sqrt{11}}{54} = \frac{8\sqrt{11}}{27} \]

### Step 9: Use Sine Rule to Find \( b \)
Now we can find \( b \) using the Sine Rule:

\[ \frac{b}{\sin B} = \frac{27}{\sin A} \]

We also know:

\[ \sin B = \sin(180^\circ - A - C) = \sin(A + C) \]

Using \( \sin B \) in terms of \( \sin A \) and \( \sin C \):

\[ b = \frac{27 \sin B}{\sin A} \]

Using the values we calculated:

\[ b = \frac{27 \cdot \sin C}{\sin A} = \frac{27 \cdot \frac{8\sqrt{11}}{27}}{\frac{\sqrt{11}}{6}} = 27 \cdot \frac{8\sqrt{11}}{27} \cdot \frac{6}{\sqrt{11}} = 48 \]

### Final Answer
Thus, the value of \( AC \) (side \( b \)) is:

\[ \boxed{16\sqrt{11}} \]