To determine the measure of the central angle for a sector with an area of [tex]\(\frac{51 \pi}{20}\)[/tex], follow these steps:
1. Understand the formula for the area of a sector:
The formula for the area [tex]\(A\)[/tex] of a sector of a circle is given by:
[tex]\[
A = \frac{\theta}{360} \cdot \pi \cdot r^2
\][/tex]
where [tex]\(\theta\)[/tex] is the central angle in degrees, and [tex]\(r\)[/tex] is the radius of the circle.
2. Identify the given values:
- Sector area [tex]\(A = \frac{51 \pi}{20}\)[/tex]
- Radius [tex]\(r = 1\)[/tex]
3. Plug the given values into the sector area formula:
Using [tex]\(A\)[/tex] and [tex]\(r\)[/tex], we write:
[tex]\[
\frac{51 \pi}{20} = \frac{\theta}{360} \cdot \pi \cdot (1)^2
\][/tex]
4. Simplify the equation:
Since the radius [tex]\(r = 1\)[/tex], the formula reduces to:
[tex]\[
\frac{51 \pi}{20} = \frac{\theta \cdot \pi}{360}
\][/tex]
5. Solve for [tex]\(\theta\)[/tex]:
- Remove [tex]\(\pi\)[/tex] from both sides of the equation:
[tex]\[
\frac{51}{20} = \frac{\theta}{360}
\][/tex]
- To isolate [tex]\(\theta\)[/tex], multiply both sides by 360:
[tex]\[
\theta = \left(\frac{51}{20}\right) \cdot 360
\][/tex]
6. Calculate [tex]\(\theta\)[/tex]:
[tex]\[
\theta = 51 \cdot \frac{360}{20}
\][/tex]
[tex]\[
\theta = 51 \cdot 18
\][/tex]
[tex]\[
\theta = 918
\][/tex]
Thus, the measure of the central angle for turn B is [tex]\(\boxed{918}\)[/tex].
