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 $\pi$.