Exercise 93 Page 466

Practice makes perfect

a To calculate the arc length of a sector, we multiply the circle's circumference with the ratio of the arc's central angle to 360^(∘).

Arc length=2π r * θ/360^(∘) By substituting the central angle and our radius in our formula, we can determine the arc length.

Arc length=2π r * θ/360^(∘)

SubstituteII

r= 5, θ= 120^(∘)

Arc length=2π ( 5) * 120^(∘)/360^(∘)

▼

Evaluate right-hand side

ReduceFrac

a/b=.a /120^(∘)./.b /120^(∘).

Arc length=2π(5) * 1/3

Multiply

Arc length=10π * 1/3

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a* 1/b= a/b

Arc length= 10π/3

The arc length is 10π3 cm. To calculate the area, we use the same principle as for the arc length. We multiply the area of the circle by the ratio of the arc's central angle to 360^(∘). Sector area=π r^2 * θ/360^(∘) By substituting the central angle and our radius in our formula, we can determine the sector area.

Sector area=π r^2 * θ/360^(∘)

SubstituteII

r= 5, θ= 120^(∘)

Sector area=π ( 5)^2 * 120^(∘)/360^(∘)

▼

Evaluate right-hand side

ReduceFrac

a/b=.a /120^(∘)./.b /120^(∘).

Sector area=π (5)^2 * 1/3

CalcPow

Calculate power

Sector area=π(25) * 1/3

CommutativePropMult

Commutative Property of Multiplication

Sector area=25π * 1/3

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a* 1/b= a/b

Sector area=25π/3

The sector area is 25π3 cm^2.

b Like in Part A, we will calculate the sector length and area by using the formulas from Part A. Notice that the central angle for this circle is a straight angle and that the diameter is 6 inches, which makes the radius 3 inches.

Arc length=2π r * θ/360^(∘)

SubstituteII

r= 3, θ= 180^(∘)

Arc length=2π ( 3) * 180^(∘)/360^(∘)

▼

Evaluate right-hand side

ReduceFrac

a/b=.a /180^(∘)./.b /180^(∘).

Arc length=2π(3) * 1/2

Multiply

Arc length=6π * 1/2

MoveLeftFacToNumOne