Question

If the arc of a circle is 15 meters long and the radius of the circle is 3 meters, then what is the measure of the central angle, to the nearest tenth of a degree, that intercepts that 15-meter arc?
(1 point)
• 7.9 degrees
• 282.7 degrees
353.4 degrees
• 286.5 degrees

Answer 1

To find the measure of the central angle that intercepts a given arc length in a circle, we can use the formula:

\[
\text{Arc Length} = r \cdot \theta
\]

where:
- \(\text{Arc Length}\) is the length of the arc,
- \(r\) is the radius of the circle,
- \(\theta\) is the central angle in radians.

Given that the arc length is 15 meters and the radius is 3 meters, we can rearrange the formula to solve for \(\theta\):

\[
\theta = \frac{\text{Arc Length}}{r}
\]

Substituting in the values:

\[
\theta = \frac{15}{3} = 5 \text{ radians}
\]

Now, to convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):

\[
\theta \text{ (in degrees)} = \theta \text{ (in radians)} \times \frac{180}{\pi}
\]

Substituting the value of \(\theta\):

\[
\theta = 5 \times \frac{180}{\pi} \approx 5 \times 57.2958 \approx 286.478 \text{ degrees}
\]

Rounding to the nearest tenth:

\[
\theta \approx 286.5 \text{ degrees}
\]

Therefore, the measure of the central angle that intercepts the 15-meter arc is **286.5 degrees**.