To determine the measure of the central angle [tex]\(\theta\)[/tex] for turn B, we start by using the formula for the area of a sector of a circle. The formula for the area [tex]\(A\)[/tex] of a sector with central angle [tex]\(\theta\)[/tex] is:
[tex]\[
A = \left( \frac{\theta}{360} \right) \pi r^2
\][/tex]
Given the sector area is:
[tex]\[
A = \frac{5 \pi x}{20}
\][/tex]
We will assume the radius [tex]\(r\)[/tex] of the circle is 1 for simplicity (since the problem does not specify the radius). Thus, the area of the sector simplifies and the formula for area becomes:
[tex]\[
A = \left( \frac{\theta}{360} \right) \pi (1)^2 = \left( \frac{\theta}{360} \right) \pi
\][/tex]
We can equate the two expressions for the area of the sector:
[tex]\[
\left( \frac{\theta}{360} \right) \pi = \frac{5 \pi x}{20}
\][/tex]
Next, we cancel [tex]\(\pi\)[/tex] from both sides of the equation:
[tex]\[
\frac{\theta}{360} = \frac{5 x}{20}
\][/tex]
Simplify the right-hand side of the equation:
[tex]\[
\frac{\theta}{360} = \frac{x}{4}
\][/tex]
To isolate [tex]\(\theta\)[/tex], multiply both sides of the equation by 360:
[tex]\[
\theta = 360 \times \frac{x}{4}
\][/tex]
Simplify the multiplication:
[tex]\[
\theta = 90 x
\][/tex]
Thus, the measure of the central angle [tex]\(\theta\)[/tex] for turn B depends on [tex]\(x\)[/tex]. If we assume [tex]\(x = 1\)[/tex], then:
[tex]\[
\theta = 90 \times 1 = 90
\][/tex]
Therefore, the measure of the central angle for turn B is:
[tex]\[
m / B = 90
\][/tex]
