Question - Calculating the Area of a Circular Fountain in a Square Park
Solution:
- 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.
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