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?
K=15​𝜋 sq. m
r=10 m
Give the exact answer in simplest form.

Answer 1

To find the angle measure of an arc bounding a sector of a circle when we know the area of the sector and the radius of the circle, 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

Given:
- \( A = 15\pi \) square meters
- \( r = 10 \) meters

We can substitute the known values into the formula:

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

Calculating \( (10)^2 \):

\[
(10)^2 = 100
\]

So the equation becomes:

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

This simplifies to:

\[
15\pi = 50 \theta
\]

Next, we solve for \( \theta \):

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

Simplifying this fraction:

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

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.