In `triangleABC,a=1,b=4, and angleC=30^(@)`. The length of c is

Options

A. 4.6
B. 3.6
C. 3.2
D. 2.9

Explanation

To find the length of side \( c \) in triangle \( ABC \) where \( a = 1 \), \( b = 4 \), and \( \angle C = 30^\circ \), we can use the Law of Cosines. The formula is given by:

\[ c^2 = a^2 + b^2 - 2ab \cos C \]

### Step-by-Step Solution:

1. **Identify the values**:
- \( a = 1 \)
- \( b = 4 \)
- \( \angle C = 30^\circ \)

2. **Substitute the values into the formula**:
\[ c^2 = 1^2 + 4^2 - 2 \cdot 1 \cdot 4 \cdot \cos(30^\circ) \]

3. **Calculate \( a^2 \) and \( b^2 \)**:
\[ 1^2 = 1 \]
\[ 4^2 = 16 \]

4. **Calculate \( 2ab \cos C \)**:
- First, find \( \cos(30^\circ) \):
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
- Now substitute this into the equation:
\[ 2 \cdot 1 \cdot 4 \cdot \cos(30^\circ) = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3} \]

5. **Combine all parts**:
\[ c^2 = 1 + 16 - 4\sqrt{3} \]
\[ c^2 = 17 - 4\sqrt{3} \]

6. **Calculate \( c \)**:
- To find \( c \), take the square root of \( c^2 \):
\[ c = \sqrt{17 - 4\sqrt{3}} \]

7. **Approximate the value**:
- Using a calculator:
\[ 4\sqrt{3} \approx 4 \cdot 1.732 \approx 6.928 \]
\[ 17 - 6.928 \approx 10.072 \]
\[ c \approx \sqrt{10.072} \approx 3.17 \]

### Final Answer:
Thus, the length of side \( c \) is approximately \( 3.17 \).