To calculate the area of a sector of a circle, you 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,
- \( r \) is the radius of the circle.
Given:
- \( \theta = 75^\circ \)
- \( r = 4 , \text{feet} \)
Substituting the values into the formula:
\[ A = \frac{75}{360} \times \pi \times (4)^2 \]
First, calculate \( (4)^2 \):
\[ (4)^2 = 16 \]
Now substitute this value into the equation:
\[ A = \frac{75}{360} \times \pi \times 16 \]
Next, simplify \( \frac{75}{360} \):
\[ \frac{75}{360} = \frac{75 \div 15}{360 \div 15} = \frac{5}{24} \]
Now, plug this back into the area formula:
\[ A = \frac{5}{24} \times \pi \times 16 \]
Now multiply \( \frac{5}{24} \) by \( 16 \):
\[ A = \frac{5 \times 16}{24} \times \pi = \frac{80}{24} \times \pi = \frac{10}{3} \times \pi \]
Therefore, the area is:
\[ A \approx 10.472 \pi , \text{ft}^2 \]
Now, let's approximate the value of the area:
\[ A \approx 10.472 \times 3.14159 \approx 32.882 \text{ ft}^2 \]
However, we also keep it in terms of \(\pi\):
\[ A = \frac{10}{3} \pi \text{ ft}^2 \approx 10.472 \text{ ft}^2 \]
Thus, the correct answer is:
A = 10.472 ft²