To find the area of the sector of a circle, we can use the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( A \) is the area of the sector, \( \theta \) is the angle in degrees, and \( r \) is the radius.
Given:
- \( \theta = 75^\circ \)
- \( r = 4 \) feet
Now we can plug in the values into the formula:
\[ A = \frac{75}{360} \times \pi \times (4)^2 \]
Calculating \( (4)^2 \):
\[ (4)^2 = 16 \]
Now substituting that value back into the formula:
\[ A = \frac{75}{360} \times \pi \times 16 \]
Calculating \( \frac{75}{360} \):
\[ \frac{75}{360} = \frac{5}{24} \]
Now substituting that back into \( A \):
\[ A = \frac{5}{24} \times \pi \times 16 \]
Calculating \( \frac{5 \times 16}{24} \):
\[ 5 \times 16 = 80 \] \[ \frac{80}{24} = \frac{10}{3} \]
So, the area of the sector \( A \) is:
\[ A = \frac{10}{3} \pi \]
This can be approximately calculated as:
\[ A \approx 10.472 \text{ ft}^2 \text{ (using } \pi \approx 3.14159\text{)} \]
Thus, the correct response is:
A = 10.472 ft².
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