Sure, let's break this problem down step-by-step:
1. Identify the given values:
- Arc length, \(\(s\)\), is \(\(2.5\pi\)\) units.
- Radius, \(\(r\)\), is 4 units.
2. Use the formula to find the central angle in radians:
The formula to find the central angle (θ) in radians is:
\(\[
\theta = \frac{s}{r}
\]\)
3. Substitute the given values into the formula:
\(\[
\theta = \frac{2.5\pi}{4}
\]\)
4. Simplify the fraction:
\(\[
\theta = \frac{2.5\pi}{4} = \frac{5\pi}{8} \text{ radians}
\]\)
So, the measure of the central angle in radians is \(\(\frac{5\pi}{8}\)\) radians.
5. Convert the central angle from radians to degrees:
To convert radians to degrees, we use the conversion factor \(\(\frac{180}{\pi}\)\):
\(\[
\theta \text{ (in degrees)} = \theta \text{ (in radians)} \times \frac{180}{\pi}
\]\)
6. Apply the conversion factor:
\(\[
\theta \text{ (in degrees)} = \frac{5\pi}{8} \times \frac{180}{\pi}
\]\)
7. Simplify and perform the calculations:
\(\[
\theta \text{ (in degrees)} = \frac{5 \times 180}{8}
\]\)
\(\[
\theta \text{ (in degrees)} = \frac{900}{8} = 112.5 \text{ degrees}
\]\)
Thus, the measure of the central angle associated with the arc is \(\( \frac{5\pi}{8} \)\) radians or 112.5 degrees.
