Question

The radius of a circle is 10 meters. What is the angle measure of an arc bounding a sector with area 15​𝜋 square meters?
Give the exact answer in simplest form.

Answer 1

To find the angle measure of an arc that bounds a sector with an area of 15π square meters in a circle of radius 10 meters, 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,
- \( \theta \) is the angle in radians.

In your case, the area \( A \) is \( 15\pi \) square meters and the radius \( r \) is 10 meters. Plugging these values into the formula gives us:

\[
15\pi = \frac{1}{2} (10)^2 \theta
\]

Simplifying the right side, we have:

\[
15\pi = \frac{1}{2} \cdot 100 \cdot \theta
\]

\[
15\pi = 50\theta
\]

Now, we can solve for \( \theta \):

\[
\theta = \frac{15\pi}{50} = \frac{3\pi}{10}
\]

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

\[
\theta = \frac{3\pi}{10} \text{ radians}
\]

This is the exact answer in simplest form.