Calculating the Area of a Circular Fountain in a Square Park
- Let the side of the square park be \( s \). Then the diagonal of the square park is \( s\sqrt{2} \).
- Given the diagonal \( s\sqrt{2} = 40 \) meters, we can find the side of the square park \( s \) by dividing the diagonal by \( \sqrt{2} \).
- So, \( s = \frac{40}{\sqrt{2}} = 20\sqrt{2} \) meters.
- The shortest distance from the edge of the fountain to each side of the park is 10 meters, which means the radius \( r \) of the circular fountain is \( r = \frac{s}{2} - 10 \).
- We plug in \( s \) to get \( r = \frac{20\sqrt{2}}{2} - 10 = 10\sqrt{2} - 10 \).
- The area \( A \) of the circular fountain is \( A = \pi r^2 \).
- Substituting the value of \( r \) we found, \( A = \pi (10\sqrt{2} - 10)^2 \).
- Expanding and applying the difference of squares formula yields \( A = \pi (200 - 200\sqrt{2} + 100) \).
- The exact area of the fountain is \( A = 300\pi - 200\sqrt{2}\pi \).
- The approximate value for \( \pi \) is 3.14159, and \( \sqrt{2} \) is approximately 1.41421.
- Thus, we approximate \( A \) as \( A \approx 300(3.14159) - 200(1.41421)(3.14159) \).
- Perform the calculations: \( A \approx 942.477 - 888.574 \).
- \( A \approx 942.477 \) square meters.
- None of the provided answer choices match the calculated area. There must be an error either in the provided answer choices or in the interpretation of the problem.
