Alright, let's go through the problem step-by-step.
### Part I: Find the central angle of one sector
First, we need to determine the central angle of one sector. We know that a full circle is [tex]\(360^\circ\)[/tex] and that the circle is divided into 20 congruent sectors. To find the angle of one sector, we divide the total angle by the number of sectors:
[tex]\[
\text{Central Angle} = \frac{360^\circ}{20}
\][/tex]
This results in:
[tex]\[
\text{Central Angle} = 18^\circ
\][/tex]
### Part II: Find the fraction of the circle that one sector occupies
Next, we need to determine the fraction of the circle that one sector represents. This can be done by dividing the central angle of one sector by the total angle of the circle (which is [tex]\(360^\circ\)[/tex]):
[tex]\[
\text{Fraction of Circle} = \frac{\text{Central Angle}}{360^\circ} = \frac{18^\circ}{360^\circ} = 0.05
\][/tex]
### Calculate the area of the sector
Finally, we need to calculate the area of one sector. The provided hint formula for the area of a sector is:
[tex]\[
\text{Area of Sector} = \frac{m \overparen{AB}}{360^\circ} \cdot \pi r^2
\][/tex]
First, we find the radius of the dartboard. Given that the diameter is 20 inches:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{20 \text{ in}}{2} = 10 \text{ in}
\][/tex]
The area of the entire circle (dartboard) is:
[tex]\[
\text{Area of Circle} = \pi r^2 = \pi (10 \text{ in})^2 = 314.1592653589793 \text{ in}^2
\][/tex]
Using the fraction of the circle we found in Part II (which is 0.05), we can find the area of one sector:
[tex]\[
\text{Area of One Sector} = \text{Fraction of Circle} \times \text{Area of Circle}
\][/tex]
Substitute the values:
[tex]\[
\text{Area of One Sector} = 0.05 \times 314.1592653589793 \text{ in}^2 = 15.707963267948967 \text{ in}^2
\][/tex]
So, the area of one sector of the dartboard is [tex]\(\approx 15.71 \text{ in}^2\)[/tex].
