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The problem presented involves calculating the area of a shaded region, where a circle is inscribed in a square. It seems that Brian made an error in his calculations. Let's carefully examine the steps and point out his mistake.

### Step-by-Step Analysis:

1. **Square Area Calculation (Correct)**:
   - The side length of the square is 10 ft (as mentioned in the image).
   - The area of the square is correctly calculated as:
     \[
     A_{\text{square}} = 10 \times 10 = 100 \, \text{ft}^2
     \]

2. **Circle Area Calculation (Correct)**:
   - The circle is inscribed in the square, so its diameter is equal to the side length of the square, i.e., 10 ft. Hence, the radius \(r\) is:
     \[
     r = \frac{10}{2} = 5 \, \text{ft}
     \]
   - The area of the circle is:
     \[
     A_{\text{circle}} = \pi \times (5)^2 = 25\pi \, \text{ft}^2
     \]

3. **Shaded Region Calculation (Brian's Mistake)**:
   - Brian subtracts the area of the circle from the square, but he incorrectly represents the difference. He writes:
     \[
     A_{\text{shaded}} = 100 - 25\pi = 75\pi
     \]
   - The error here is in the final expression. He mistakenly subtracted the areas but retained the \(\pi\) factor in the result, which is incorrect.

### Correct Calculation for the Shaded Area:
- The correct way to calculate the shaded area is:
  \[
  A_{\text{shaded}} = A_{\text{square}} - A_{\text{circle}} = 100 - 25\pi
  \]
  - This expression cannot be simplified further unless you approximate \(\pi\).
  - Using \(\pi \approx 3.1416\), we get:
    \[
    A_{\text{shaded}} = 100 - 25 \times 3.1416 = 100 - 78.54 = 21.46 \, \text{ft}^2
    \]

### Final Answer:
The true area of the shaded region is approximately **21.46 ft²**.

Would you like further details or have any questions?

### Related Questions:
1. How would the area change if the side of the square were doubled?
2. What would happen to the shaded area if the circle had a radius of 6 ft?
3. How can the same method be used to find the shaded area if the shape inscribed was a triangle instead of a circle?
4. How is the concept of area subtraction used in more complex geometry problems?
5. What role does the value of \(\pi\) play in determining the accuracy of area measurements?

### Tip:
Always double-check your units and factors when subtracting areas that involve constants like \(\pi\).

Brian was attempting to calculate the exact area of the shaded region, where a circle is inscribed in a square. He made an error in his calculation and found the shaded area as 75π. What was his mistake, and what is the correct shaded area to the nearest hundredth?

In this problem, a circle is inscribed in a square, and Brian mistakenly calculates the shaded region's area as 75π. Explore the correct step-by-step process to find the exact area, using the formulas for the area of a square and a circle, and subtracting the areas. Appropriate for Grades 7-9.

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