Question
What is the area of the sector of a circle with θ=75°
and a radius of 4 feet? Include pi in your calculation.(1 point)
Responses
A=10.472 ft.2
upper A equals 10.472 ft. squared
A=2.168 ft.2
upper A equals 2.168 ft. squared
A=39.794 ft.2
upper A equals 39.794 ft. squared
A=50.265 ft.2
and a radius of 4 feet? Include pi in your calculation.(1 point)
Responses
A=10.472 ft.2
upper A equals 10.472 ft. squared
A=2.168 ft.2
upper A equals 2.168 ft. squared
A=39.794 ft.2
upper A equals 39.794 ft. squared
A=50.265 ft.2
Answers
GPT-4o mini
To find the area of a sector of a circle, you can use the formula:
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( A \) is the area of the sector, \( \theta \) is the angle in degrees, and \( r \) is the radius of the circle.
In your case, the radius \( r = 4 \) feet and the angle \( \theta = 75^\circ \).
Substituting the values into the formula:
\[
A = \frac{75}{360} \times \pi \times (4)^2
\]
Calculating this step-by-step:
1. Calculate \( (4)^2 = 16 \).
2. Calculate \( \frac{75}{360} = \frac{75 \div 15}{360 \div 15} = \frac{5}{24} \).
3. Now substitute these values back into the formula:
\[
A = \frac{5}{24} \times \pi \times 16
\]
Calculating further:
\[
A = \frac{5 \times 16}{24} \times \pi = \frac{80}{24} \times \pi = \frac{10}{3} \times \pi
\]
To get a numerical approximation of the area:
\[
A \approx (3.3333) \times \pi \approx 10.472 \text{ ft}^2
\]
Thus, the area of the sector is:
\[
\text{Area} \approx 10.472 \text{ ft}^2
\]
This corresponds to the first response option.
So, the correct answer is:
**A=10.472 ft.²**
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \( A \) is the area of the sector, \( \theta \) is the angle in degrees, and \( r \) is the radius of the circle.
In your case, the radius \( r = 4 \) feet and the angle \( \theta = 75^\circ \).
Substituting the values into the formula:
\[
A = \frac{75}{360} \times \pi \times (4)^2
\]
Calculating this step-by-step:
1. Calculate \( (4)^2 = 16 \).
2. Calculate \( \frac{75}{360} = \frac{75 \div 15}{360 \div 15} = \frac{5}{24} \).
3. Now substitute these values back into the formula:
\[
A = \frac{5}{24} \times \pi \times 16
\]
Calculating further:
\[
A = \frac{5 \times 16}{24} \times \pi = \frac{80}{24} \times \pi = \frac{10}{3} \times \pi
\]
To get a numerical approximation of the area:
\[
A \approx (3.3333) \times \pi \approx 10.472 \text{ ft}^2
\]
Thus, the area of the sector is:
\[
\text{Area} \approx 10.472 \text{ ft}^2
\]
This corresponds to the first response option.
So, the correct answer is:
**A=10.472 ft.²**