The length of a chain used as the boundary of a semicircular park is `108m`. Find the area of the park.

Explanation

To solve the problem of finding the area of a semicircular park given that the length of the chain used as the boundary is 108 meters, we can follow these steps:

### Step 1: Understand the boundary of the semicircular park
The boundary of a semicircular park consists of half the circumference of the circle plus the diameter. The formula for the circumference of a circle is \(C = 2\pi r\). Therefore, the boundary length \(L\) for a semicircle can be expressed as:
\[ L = \frac{1}{2} \times 2\pi r + d \]
where \(d\) is the diameter of the semicircle, which is equal to \(2r\). Thus, we can write:
\[ L = \pi r + 2r \]

### Step 2: Set up the equation
Given that the length of the chain (the boundary) is 108 meters, we can set up the equation:
\[ \pi r + 2r = 108 \]

### Step 3: Simplify the equation
We can factor out \(r\) from the left side:
\[ r(\pi + 2) = 108 \]

### Step 4: Solve for \(r\)
To find \(r\), we can rearrange the equation:
\[ r = \frac{108}{\pi + 2} \]
Using \(\pi \approx \frac{22}{7}\), we substitute:
\[ r = \frac{108}{\frac{22}{7} + 2} \]
Converting 2 to a fraction:
\[ r = \frac{108}{\frac{22}{7} + \frac{14}{7}} = \frac{108}{\frac{36}{7}} = 108 \times \frac{7}{36} \]
Now, simplifying:
\[ r = \frac{756}{36} = 21 \text{ meters} \]

### Step 5: 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 \(r = 21\) meters:
\[ A = \frac{1}{2} \times \frac{22}{7} \times (21)^2 \]
Calculating \(21^2\):
\[ 21^2 = 441 \]
Now substituting back:
\[ A = \frac{1}{2} \times \frac{22}{7} \times 441 \]
Calculating further:
\[ A = \frac{22 \times 441}{14} = \frac{9702}{14} = 693 \text{ square meters} \]

### Final Answer:
The area of the semicircular park is \(693 \text{ square meters}\).

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