To find the measure of the central angle that intercepts a given arc, you 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.
In this case, the arc length is 15 meters and the radius \(r\) is 3 meters. We need to solve for \(\theta\):
\[ 15 = 3 \cdot \theta \]
Dividing both sides by 3 gives:
\[ \theta = \frac{15}{3} = 5 \text{ radians} \]
Next, we need to convert \(\theta\) from radians to degrees. The conversion factor is:
\[ 180^\circ = \pi \text{ radians} \]
Thus:
\[ \theta \text{ in degrees} = \theta \text{ in radians} \cdot \frac{180^\circ}{\pi} \]
Substituting in our value for \(\theta\):
\[ \theta \text{ in degrees} = 5 \cdot \frac{180^\circ}{\pi} \approx 5 \cdot 57.2958 \approx 286.5^\circ \]
So the central angle that intercepts the 15-meter arc is approximately \(286.5\) degrees.
The correct response is:
286.5 degrees