The perimeter of a sheet of paper in the shape of a quadrant of a circle is 75 cm. Find its area

Answer

To solve the problem, we need to find the area of a quadrant of a circle given that its perimeter is 75 cm.

### Step-by-Step Solution:

1. **Understand the Shape**:
The sheet of paper is in the shape of a quadrant of a circle. A quadrant is one-fourth of a circle.

2. **Identify the Perimeter Formula**:
The perimeter \( P \) of a quadrant of a circle is given by the formula:
\[ P = 2R + \frac{1}{2} \pi R \]
where \( R \) is the radius of the circle.

3. **Set Up the Equation**:
We know the perimeter is 75 cm, so we can set up the equation:
\[ 75 = 2R + \frac{1}{2} \pi R \]

4. **Substitute the Value of \( \pi \)**:
Using \( \pi \approx \frac{22}{7} \), we substitute this into the equation:
\[ 75 = 2R + \frac{1}{2} \cdot \frac{22}{7} R \]

5. **Simplify the Equation**:
First, calculate \( \frac{1}{2} \cdot \frac{22}{7} \):
\[ \frac{1}{2} \cdot \frac{22}{7} = \frac{22}{14} = \frac{11}{7} \]
Now substitute this back into the equation:
\[ 75 = 2R + \frac{11}{7} R \]

6. **Combine Like Terms**:
To combine \( 2R \) and \( \frac{11}{7} R \), convert \( 2R \) into a fraction with a denominator of 7:
\[ 2R = \frac{14}{7} R \]
Now we can combine:
\[ 75 = \left(\frac{14}{7} + \frac{11}{7}\right) R = \frac{25}{7} R \]

7. **Solve for \( R \)**:
Multiply both sides by 7 to eliminate the fraction:
\[ 75 \cdot 7 = 25R \]
\[ 525 = 25R \]
Now divide by 25:
\[ R = \frac{525}{25} = 21 \text{ cm} \]

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

9. **Simplify the Area Calculation**:
First, calculate \( \frac{22 \cdot 441}{7} \):
\[ \frac{22 \cdot 441}{7} = \frac{9702}{7} = 1386 \]
Now divide by 4:
\[ A = \frac{1386}{4} = 346.5 \text{ cm}^2 \]

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