Turn B has a sector area of [tex]\frac{51 \pi}{20}[/tex]. What is the measure of the central angle for turn B?

Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.

[tex]
m \angle B = \square
[/tex]



Answer :

Sure, let's walk through the solution.

We are given the sector area of turn B, which is [tex]\(\frac{51 \pi}{20}\)[/tex].

The formula for the area of a sector of a circle is:

[tex]\[ \text{Sector area} = \frac{\theta}{360} \times \pi r^2 \][/tex]

where [tex]\(\theta\)[/tex] is the central angle in degrees and [tex]\(r\)[/tex] is the radius of the circle. Assuming the radius [tex]\(r\)[/tex] is 1 (unit circle), the formula simplifies to:

[tex]\[ \text{Sector area} = \frac{\theta}{360} \times \pi \][/tex]

Given the sector area, we can set up the equation:

[tex]\[ \frac{\theta}{360} \times \pi = \frac{51 \pi}{20} \][/tex]

To solve for [tex]\(\theta\)[/tex], we first isolate it on one side of the equation.

1. Eliminate [tex]\(\pi\)[/tex] by dividing both sides by [tex]\(\pi\)[/tex]:

[tex]\[ \frac{\theta}{360} = \frac{51}{20} \][/tex]

2. Next, solve for [tex]\(\theta\)[/tex] by multiplying both sides by 360:

[tex]\[ \theta = \frac{51}{20} \times 360 \][/tex]

3. Perform the multiplication:

[tex]\[ \theta = 51 \times 18 \][/tex]
[tex]\[ \theta = 918 \][/tex]

Thus, the measure of the central angle for turn B is:

[tex]\[ m \angle B = 918 \][/tex]

So the correct answer is:
[tex]\[ \boxed{918} \][/tex]