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.

a. A = 10.472 ft.^2
b. A = 39.794 ft.^2
c. A = 50.265 ft.^2
d. A = 2.168 ft.^2

Answer 1

To find the area \( A \) of a sector of a circle, you can use the formula:

\[
A = \frac{θ}{360} \times πr^2
\]

where \( θ \) is the angle in degrees and \( r \) is the radius.

In this case, the angle \( θ = 75^\circ \) and the radius \( r = 4 \) feet. Plugging these values into the formula:

\[
A = \frac{75}{360} \times π \times (4)^2
\]

First, calculate \( (4)^2 \):

\[
(4)^2 = 16
\]

Next, substitute this back into the formula:

\[
A = \frac{75}{360} \times π \times 16
\]

Now simplify \( \frac{75}{360} \):

\[
\frac{75}{360} = \frac{75 \div 15}{360 \div 15} = \frac{5}{24}
\]

Now substitute this into the equation:

\[
A = \frac{5}{24} \times π \times 16
\]

Now multiply \( \frac{5 \times 16}{24} \):

\[
\frac{5 \times 16}{24} = \frac{80}{24}
\]

Now simplify \( \frac{80}{24} \):

\[
\frac{80}{24} = \frac{80 \div 8}{24 \div 8} = \frac{10}{3}
\]

So, we have:

\[
A = \frac{10}{3} π
\]

To find the approximate numerical value:

\[
A \approx \frac{10}{3} \times 3.14159 \approx 10.472
\]

Thus, the area of the sector is approximately \( 10.472 \) ft².

The correct answer is:

**a. A = 10.472 ft²**