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