The perimeter of a sheet of tin in the shape a quadrant of a circle is 12.5 cm. Find its area

Answer

To solve the problem step by step, we need to find the area of a quadrant of a circle given its perimeter.

### Step 1: Understand the Perimeter of the Quadrant
The perimeter \( P \) of a quadrant of a circle consists of two radii and one-fourth of the circumference of the circle. The formula for the perimeter is:
\[ P = 2r + \frac{1}{4} \times 2\pi r = 2r + \frac{\pi r}{2} \]

### Step 2: Set Up the Equation
We know from the problem that the perimeter is given as 12.5 cm. Therefore, we can set up the equation:
\[ 2r + \frac{\pi r}{2} = 12.5 \]

### Step 3: Substitute the Value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\), we can substitute this value into the equation:
\[ 2r + \frac{22}{7} \cdot \frac{r}{2} = 12.5 \]

### Step 4: Simplify the Equation
Now, let's simplify the equation:
\[ 2r + \frac{11r}{7} = 12.5 \]
To eliminate the fraction, we can multiply the entire equation by 7:
\[ 7(2r) + 11r = 7 \cdot 12.5 \]
This simplifies to:
\[ 14r + 11r = 87.5 \]
\[ 25r = 87.5 \]

### Step 5: Solve for \( r \)
Now, divide both sides by 25 to find \( r \):
\[ r = \frac{87.5}{25} = 3.5 \text{ cm} \]

### Step 6: Calculate the Area of the Quadrant
The area \( A \) of a quadrant of a circle is given by:
\[ A = \frac{1}{4} \times \pi r^2 \]
Substituting \( r = 3.5 \) cm and \(\pi \approx \frac{22}{7}\):
\[ A = \frac{1}{4} \times \frac{22}{7} \times (3.5)^2 \]

### Step 7: Calculate \( (3.5)^2 \)
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]

### Step 8: Substitute and Simplify
Now substitute back into the area formula:
\[ A = \frac{1}{4} \times \frac{22}{7} \times 12.25 \]
Calculating:
\[ A = \frac{22 \times 12.25}{28} = \frac{270.5}{28} = 9.625 \text{ cm}^2 \]

### Final Answer
Thus, the area of the quadrant of the circle is:
\[ \boxed{9.625 \text{ cm}^2} \]