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.
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