To find the area of the baseball field, which is in the shape of a sector of a circle, we need to use the formula for the area of a sector. The formula for the area \( A \) of a sector with radius \( r \) and central angle \( \theta \) (in radians) is:
[tex]\[
A = \frac{1}{2} \theta r^2
\][/tex]
Given parameters:
- Radius (\( r \)) = 300 feet
- Central angle (\( \theta \)) = 90 degrees
First, we need to convert the angle from degrees to radians, because the formula requires the angle in radians. Recall that \( \pi \) radians = 180 degrees. Therefore, to convert 90 degrees to radians, we use:
[tex]\[
\theta = 90 \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians}
\][/tex]
Now we substitute the known values into the area formula:
[tex]\[
A = \frac{1}{2} \times \frac{\pi}{2} \times (300)^2
\][/tex]
We know that \( \pi \approx 3.14 \), so:
[tex]\[
\theta \approx \frac{3.14}{2} = 1.57 \text{ radians}
\][/tex]
Substituting this value into our formula, we get:
[tex]\[
A = \frac{1}{2} \times 1.57 \times 300^2
\][/tex]
Calculating \( 300^2 \):
[tex]\[
300^2 = 90000
\][/tex]
Now, multiply the values together:
[tex]\[
A = \frac{1}{2} \times 1.57 \times 90000
\][/tex]
[tex]\[
A = 0.5 \times 1.57 \times 90000
\][/tex]
[tex]\[
A = 0.785 \times 90000
\][/tex]
[tex]\[
A \approx 70685.83470577035 \text{ square feet}
\][/tex]
Thus, the area of the baseball field is approximately 70685.83 square feet.
