The perimeter of a semicircular path is 36 m. Find the area of this semicircular path.

Options

A. 42 sq. m
B. 54 sq. m
C. 63 sq. m
D. 77 sq. m

Answer

To solve the problem, we need to find the area of a semicircular path given that its perimeter is 36 meters.

### Step-by-step Solution:

1. **Understand the Perimeter of a Semicircle**:
The perimeter \( P \) of a semicircular path consists of the curved part (which is half the circumference of a full circle) and the straight part (the diameter). The formula for the perimeter of a semicircle is:
\[ P = \frac{1}{2} \times 2\pi r + 2r = \pi r + 2r \]

2. **Set Up the Equation**:
Given that the perimeter \( P = 36 \) meters, we can set up the equation:
\[ \pi r + 2r = 36 \]

3. **Substitute the Value of \( \pi \)**:
We will use \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r + 2r = 36 \]

4. **Combine the Terms**:
To combine the terms, we need a common denominator. The common denominator here is 7:
\[ \frac{22r}{7} + \frac{14r}{7} = 36 \]
This simplifies to:
\[ \frac{36r}{7} = 36 \]

5. **Solve for \( r \)**:
Multiply both sides by 7 to eliminate the fraction:
\[ 36r = 36 \times 7 \]
\[ 36r = 252 \]
Now, divide both sides by 36:
\[ r = \frac{252}{36} = 7 \text{ meters} \]

6. **Calculate the Area of the Semicircle**:
The area \( A \) of a semicircle is given by the formula:
\[ A = \frac{1}{2} \pi r^2 \]
Substituting the value of \( r \):
\[ A = \frac{1}{2} \times \frac{22}{7} \times (7)^2 \]
\[ A = \frac{1}{2} \times \frac{22}{7} \times 49 \]
\[ A = \frac{1}{2} \times \frac{1078}{7} \]
\[ A = \frac{539}{7} = 77 \text{ square meters} \]

### Final Answer:
The area of the semicircular path is **77 square meters**.