The perimeter of a sheet of paper in the shape of a quadrant of a circle is 75 cm. Its area would be `(pi = (22)/(7))`

Options

A. `100 cm^(2)`
B. `346.5 cm^(2)`
C. `693 cm^(2)`
D. `512.25 cm^(2)`

Answer

To find the area of a sheet of paper in the shape of a quadrant of a circle, given that its perimeter is 75 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Perimeter of a Quadrant:**
The perimeter \( P \) of a quadrant of a circle consists of two radii and the arc length. The formula for the perimeter is:
\[ P = 2R + L \]
where \( R \) is the radius and \( L \) is the length of the arc.

2. **Finding the Arc Length:**
The arc length \( L \) of a quadrant is one-fourth of the circumference of the full circle. The circumference \( C \) of a circle is given by:
\[ C = 2\pi R \]
Therefore, the arc length \( L \) for a quadrant is:
\[ L = \frac{1}{4} \times C = \frac{1}{4} \times 2\pi R = \frac{\pi R}{2} \]

3. **Substituting the Arc Length into the Perimeter Formula:**
Now substituting \( L \) in the perimeter formula:
\[ P = 2R + \frac{\pi R}{2} \]
Given that the perimeter \( P \) is 75 cm, we can set up the equation:
\[ 75 = 2R + \frac{\pi R}{2} \]

4. **Substituting the Value of \(\pi\):**
We are given \(\pi = \frac{22}{7}\). Substituting this value into the equation:
\[ 75 = 2R + \frac{22}{7} \cdot \frac{R}{2} \]
Simplifying the second term:
\[ 75 = 2R + \frac{11R}{7} \]

5. **Finding a Common Denominator:**
To combine the terms, we can express \( 2R \) with a common denominator of 7:
\[ 2R = \frac{14R}{7} \]
Thus, the equation becomes:
\[ 75 = \frac{14R}{7} + \frac{11R}{7} \]
Combining the fractions:
\[ 75 = \frac{25R}{7} \]

6. **Solving for \( R \):**
Multiply both sides by 7 to eliminate the fraction:
\[ 525 = 25R \]
Dividing both sides by 25 gives:
\[ R = \frac{525}{25} = 21 \text{ cm} \]

7. **Calculating the Area of the Quadrant:**
The area \( A \) of a quadrant is given by:
\[ A = \frac{1}{4} \pi R^2 \]
Substituting the value of \( R \) and \(\pi\):
\[ A = \frac{1}{4} \cdot \frac{22}{7} \cdot (21)^2 \]
Calculating \( (21)^2 = 441 \):
\[ A = \frac{1}{4} \cdot \frac{22}{7} \cdot 441 \]

8. **Simplifying the Area Calculation:**
First, calculate \( \frac{441}{4} = 110.25 \):
\[ A = \frac{22 \cdot 110.25}{7} \]
Now calculating \( 22 \cdot 110.25 = 2425.5 \):
\[ A = \frac{2425.5}{7} = 346.5 \text{ cm}^2 \]

### Final Answer:
The area of the sheet of paper in the shape of a quadrant of a circle is \( 346.5 \text{ cm}^2 \).

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