Question

The radius of a circle is 15 feet. What is the angle measure of an arc bounding a sector with area 15​𝜋 square feet?
K=15​𝜋 sq. ft
r=15 ft
Give the exact answer in simplest form.

Answer 1

To find the angle measure of an arc bounding a sector with an area of \( 15\pi \) square feet in a circle with a radius of 15 feet, we can use the formula for the area of a sector:

\[
A = \frac{1}{2} r^2 \theta
\]

where:
- \( A \) is the area of the sector,
- \( r \) is the radius of the circle, and
- \( \theta \) is the angle in radians.

Given:
- \( A = 15\pi \) square feet
- \( r = 15 \) feet

Substituting these values into the formula, we have:

\[
15\pi = \frac{1}{2} (15^2) \theta
\]

Calculating \( 15^2 \):

\[
15^2 = 225
\]

Now substituting this back into the equation:

\[
15\pi = \frac{1}{2} (225) \theta
\]

Simplifying:

\[
15\pi = 112.5 \theta
\]

To solve for \( \theta \), we divide both sides by \( 112.5 \):

\[
\theta = \frac{15\pi}{112.5}
\]

This can be simplified further. First, simplify the fraction:

\[
\theta = \frac{15\pi}{112.5} = \frac{15\pi}{\frac{225}{2}} = \frac{15\pi \times 2}{225}
\]

This gives us:

\[
\theta = \frac{30\pi}{225}
\]

To simplify \( \frac{30\pi}{225} \):

\[
\theta = \frac{30\pi \div 15}{225 \div 15} = \frac{2\pi}{15}
\]

Thus, the angle measure of the arc bounding the sector is:

\[
\theta = \frac{2\pi}{15} \text{ radians}
\]

This is the exact answer in simplest form.