A thin wire 48 cm long, is bent to form a rectangle. The breadth of the rectangle is one-third its length. What is the area of the rectangle?

Options

A. `118 cm^(2)`
B. `102 cm^(2)`
C. `98 cm^(2)`
D. `108 cm^(2)`

Answer

To find the area of the rectangle formed by bending a thin wire of length 48 cm, where the breadth is one-third of the length, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Relationship Between Length and Breadth**:
- Let the length of the rectangle be \( L \).
- According to the problem, the breadth \( B \) is one-third of the length. Therefore, we can express this as:
\[ B = \frac{L}{3} \]

2. **Write the Perimeter Equation**:
- The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
- Since the wire is 48 cm long, we have:
\[ 2(L + B) = 48 \]

3. **Substitute the Expression for Breadth**:
- Substitute \( B = \frac{L}{3} \) into the perimeter equation:
\[ 2\left(L + \frac{L}{3}\right) = 48 \]

4. **Simplify the Equation**:
- Combine the terms inside the parentheses:
\[ L + \frac{L}{3} = \frac{3L}{3} + \frac{L}{3} = \frac{4L}{3} \]
- Now substitute this back into the perimeter equation:
\[ 2 \cdot \frac{4L}{3} = 48 \]
- This simplifies to:
\[ \frac{8L}{3} = 48 \]

5. **Solve for Length \( L \)**:
- Multiply both sides by 3 to eliminate the fraction:
\[ 8L = 144 \]
- Now divide by 8:
\[ L = 18 \text{ cm} \]

6. **Calculate the Breadth \( B \)**:
- Now that we have \( L \), we can find \( B \):
\[ B = \frac{L}{3} = \frac{18}{3} = 6 \text{ cm} \]

7. **Calculate the Area of the Rectangle**:
- The area \( A \) of a rectangle is given by:
\[ A = L \times B \]
- Substitute the values of \( L \) and \( B \):
\[ A = 18 \times 6 = 108 \text{ cm}^2 \]

### Final Answer:
The area of the rectangle is \( 108 \text{ cm}^2 \).