   Chapter 3.7, Problem 39E

Chapter
Section
Textbook Problem

# If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 32 inches, what diameter pizza will reward you with the largest slice?

To determine

To find:

Diameter of a pizza that will reward you with the largest slice

Explanation

1) Concept:

Use the formula for area of a sector of a circle to calculate the diameter of the pizza

2) Formula:

i) Arc length l=θ3602πr

ii) Perimeter of a sector is Arc length + Radius + Radius

iii) Area of sector of circle is 12θ·r2

3) Given:

Perimeter of slice is 32 inches

4) Calculation:

Let r be the radius of circle.

By using formula i) in ii),

=θ360°360°r+r+r=rθ+2r

Perimeter of slice =rθ+2r

32=rθ+2r(1)

32=rθ+2r

Subtracting 2r on both sides,

32-2r=rθ

Divide by r on both sides,

32-2rr=θ

To find the diameter of pizza that will reward you with the largest slice, the area of pizza should be maximum

Area of slice of pizza is A=12θ·r2

Substitute value of θ

A(r)=1232-2rrr2

Ar=1232-2rrr2

Factoring out 2 common,

Ar=2216-rr

Cancelli

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