Question

The radius of a circle is 20 miles. What is the angle measure of an arc bounding a sector with area 16​𝜋 square miles?
K=16​𝜋 sq. mi
r=20 mi
Give the exact answer in simplest form.

Answer 1

To find the angle measure of an arc bounding a sector with a given area, we can use the formula for the area of a sector of a circle:

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

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

Given:
- \( A = 16\pi \, \text{sq. mi} \)
- \( r = 20 \, \text{mi} \)

We can plug the values into the formula:

\[
16\pi = \frac{1}{2} (20^2) \theta
\]

Calculating \( 20^2 \):

\[
20^2 = 400
\]

Substituting this back into the equation:

\[
16\pi = \frac{1}{2} \cdot 400 \cdot \theta
\]

This simplifies to:

\[
16\pi = 200\theta
\]

Now, solving for \( \theta \):

\[
\theta = \frac{16\pi}{200}
\]

Simplifying this fraction:

\[
\theta = \frac{16\pi}{200} = \frac{16\pi}{100} = \frac{8\pi}{50} = \frac{4\pi}{25}
\]

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

\[
\theta = \frac{4\pi}{25} \text{ radians}
\]

This is the exact answer in simplest form.