The perimeter of the quadrant of a circle is 37.5 cm. Find area of the circle (in `cm^(2)`).

Options

A. 346.5
B. 325.5
C. 275.25
D. 173.25

Explanation

To find the area of the circle from the given perimeter of the quadrant, we can follow these steps:

### Step 1: Understand the Perimeter of the Quadrant
The perimeter of a quadrant of a circle consists of two radii and the arc length. If we denote the radius of the circle as \( r \), the perimeter \( P \) can be expressed as:
\[ P = r + r + \text{Arc Length} \]
The arc length for a 90-degree angle (which is a quadrant) is given by:
\[ \text{Arc Length} = \frac{90}{360} \times 2\pi r = \frac{\pi r}{2} \]
Thus, the total perimeter becomes:
\[ P = r + r + \frac{\pi r}{2} = 2r + \frac{\pi r}{2} \]

### Step 2: Set Up the Equation
Given that the perimeter of the quadrant is \( 37.5 \, \text{cm} \), we can set up the equation:
\[ 2r + \frac{\pi r}{2} = 37.5 \]

### Step 3: Solve for \( r \)
To solve for \( r \), we first combine the terms:
\[ 2r + \frac{\pi r}{2} = 37.5 \]
To eliminate the fraction, multiply the entire equation by 2:
\[ 4r + \pi r = 75 \]
Now, factor out \( r \):
\[ r(4 + \pi) = 75 \]
Now, solve for \( r \):
\[ r = \frac{75}{4 + \pi} \]

### Step 4: Calculate \( r \)
Using \( \pi \approx 3.14 \):
\[ r \approx \frac{75}{4 + 3.14} = \frac{75}{7.14} \approx 10.52 \, \text{cm} \]

### Step 5: Find the Area of the Circle
The area \( A \) of the circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A \approx \pi (10.52)^2 \]
Calculating \( (10.52)^2 \):
\[ (10.52)^2 \approx 110.70 \]
Now substituting back into the area formula:
\[ A \approx 3.14 \times 110.70 \approx 348.80 \, \text{cm}^2 \]

### Step 6: Final Answer
The area of the circle is approximately:
\[ \boxed{348.80 \, \text{cm}^2} \]

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