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6. A sector of a circle is a region bound by an arc and the two radii that share the arc's endpoints. Suppose you have a dartboard that has a diameter of 20 in and it is divided into 20 congruent sectors. Find the area of one sector.

The area of a sector is given by:
[tex]\[ \text{Area of sector} = \frac{m \overline{AB}}{360^\circ} \cdot \pi r^2 \][/tex]

Part I: Find the central angle.
(Hint: A circle has [tex]\(360^\circ\)[/tex].) (1 point)
[tex]\[ \frac{360^\circ}{20} = 18^\circ \][/tex]

Part II: Use your answer from Part I to find the fraction of the circle that one sector will take up. (1 point)
[tex]\[ \frac{18^\circ}{360^\circ} = \frac{1}{20} \][/tex]

Part III: Use the fractional part from Part II with the area formula to find the area of one sector of the circle to the nearest tenth. (2 points)
[tex]\[
\text{Area of the circle} = \pi (10)^2 = 314.2 \, \text{in}^2 \\
\text{Area of one sector} = \frac{1}{20} \times 314.2 \, \text{in}^2 = 15.7 \, \text{in}^2
\][/tex]



Answer :

GinnyAnswer
Let's solve the problem step-by-step.

### Part I: Find the Central Angle

A circle has a total of 360 degrees. Since the circle is divided into 20 equal sectors, the central angle of each sector can be found using:

[tex]\[ \text{Central Angle} = \frac{360^\circ}{\text{Number of Sectors}} \][/tex]

Given:
- Number of Sectors = 20

So,

[tex]\[ \text{Central Angle} = \frac{360^\circ}{20} = 18^\circ \][/tex]

### Part II: Find the Fraction of the Circle that One Sector Will Take Up

The fraction of the circle that one sector will occupy can be determined using the ratio of the central angle to the total degrees in a circle.

[tex]\[ \text{Fraction of Circle} = \frac{\text{Central Angle}}{360^\circ} \][/tex]

Substituting the central angle found in Part I:

[tex]\[ \text{Fraction of Circle} = \frac{18^\circ}{360^\circ} = 0.05 \][/tex]

### Part III: Calculate the Area of One Sector

To find the area of one sector, we'll first calculate the area of the entire circle and then find the portion of this area corresponding to one sector.

#### Step 1: Find the Area of the Entire Circle

Given:
- Diameter of the circle, [tex]\( d = 20 \)[/tex] inches

We can find the radius [tex]\( r \)[/tex] by dividing the diameter by 2:

[tex]\[ r = \frac{d}{2} = \frac{20}{2} = 10 \text{ inches} \][/tex]

The area of a circle is given by:

[tex]\[ \text{Area of Circle} = \pi r^2 \][/tex]

Substituting the radius:

[tex]\[ \text{Area of Circle} = \pi \times (10)^2 = 100\pi \approx 314.1593 \text{ square inches} \][/tex]

#### Step 2: Find the Area of One Sector

We use the fraction of the circle found in Part II to determine the area of one sector.

[tex]\[ \text{Area of Sector} = \text{Fraction of Circle} \times \text{Area of Circle} \][/tex]
[tex]\[ \text{Area of Sector} = 0.05 \times 314.1593 \text{ square inches} \approx 15.708 \text{ square inches} \][/tex]

To the nearest tenth, the area of one sector is:

[tex]\[ \boxed{15.7 \text{ square inches}} \][/tex]

So, one sector of your dartboard, with a diameter of 20 inches divided into 20 equal parts, has an area of approximately 15.7 square inches.

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