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 step by step, we will start by understanding the shape and its properties.

### Step 1: Understand the shape
The shape is a quadrant of a circle, which means it is a quarter of a full circle. The perimeter of this quadrant consists of two straight edges (the radius) and the curved edge (the arc).

### Step 2: Write the formula for the perimeter
The perimeter \( P \) of a quadrant of a circle can be expressed as:
\[ P = \frac{1}{4} \times 2\pi r + 2r \]
This simplifies to:
\[ P = \frac{\pi r}{2} + 2r \]

### Step 3: Set up the equation
We know from the problem that the perimeter is 75 cm. Therefore, we can set up the equation:
\[ \frac{\pi r}{2} + 2r = 75 \]

### Step 4: Substitute the value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\), we substitute this into the equation:
\[ \frac{22}{7} \cdot \frac{r}{2} + 2r = 75 \]

### Step 5: Clear the fraction
Multiply the entire equation by 14 (the least common multiple of 7 and 2) to eliminate the fraction:
\[ 14 \left(\frac{22}{7} \cdot \frac{r}{2}\right) + 14 \cdot 2r = 14 \cdot 75 \]
This simplifies to:
\[ 22r + 28r = 1050 \]

### Step 6: Combine like terms
Combine the terms on the left side:
\[ 50r = 1050 \]

### Step 7: Solve for \( r \)
Now, divide both sides by 50 to find \( r \):
\[ r = \frac{1050}{50} = 21 \text{ cm} \]

### Step 8: Find the area of the quadrant
The area \( A \) of a quadrant of a circle is given by:
\[ A = \frac{1}{4} \pi r^2 \]
Substituting \( r = 21 \) cm and \(\pi \approx \frac{22}{7}\):
\[ A = \frac{1}{4} \cdot \frac{22}{7} \cdot (21)^2 \]

### Step 9: Calculate \( (21)^2 \)
Calculating \( (21)^2 \):
\[ (21)^2 = 441 \]

### Step 10: Substitute and calculate the area
Now substitute back into the area formula:
\[ A = \frac{1}{4} \cdot \frac{22}{7} \cdot 441 \]
Calculating:
\[ A = \frac{22 \cdot 441}{28} \]
\[ A = \frac{9702}{28} = 346.5 \text{ cm}^2 \]

### Final Answer
The area of the quadrant of the circle is \( 346.5 \text{ cm}^2 \).

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