Exercise 52 Page 656

The length of an arc of a circle is the product of the measure of the arc divided by 360 and the circumference of the circle. length of AB=mAB/360 * 2π r Therefore, to find the length of the desired arc, we first need to find its measure. For simplicity, let's label the endpoints of the involved arcs.

Looking at the diagram, we can see that the desired ACD is formed by the adjacent arcs AB and the semicircle BCD.

By the Arc Addition Postulate, we know that the m ACD is the sum of the measures of these two adjacent arcs. m ACD=m AB+mBCD From the diagram, we know that m AB is 45^(∘). Also, note that BCD is a semicircle, therefore, its measure is 180^(∘). Let's substitute this value in the above equation, and solve for m ACD.

mACD=mAB+mBCD

SubstituteII

mAB= 45, mBCD= 180

mACD= 45+180

AddTerms

Add terms

mACD=225

Therefore, the measure of the arc ACD is 225.

We can also see that the radius of the circle is 4.1 feet. Let's substitute m ACD= 225 and r=4.1 in the formula for arc length and simplify.

Length of ACD=mACD/360 * 2π r

SubstituteII

mACD= 225, r= 4.1

Length of ACD=225/360 * 2π (4.1)

▼

Evaluate right-hand side

CommutativePropMult

Commutative Property of Multiplication

Length of ACD=225/360 * 2(4.1)π

Multiply

Multiply

Length of ACD=225/360 * (8.2)π

MoveRightFacToNum

a/c* b = a* b/c

Length of ACD=1845π/360

ReduceFrac

a/b=.a /45./.b /45.

Length of ACD=41π/8

The length of the arc is 41π8 feet.