Question

If you are offered one slice from a round pizza and the slice must have a perimeter of 36 inches, what diameter pizza will reward you with the largest slice

Answer 1

A diameter of 16 inches will give the largest slice

Explanation:

Step 1: Statement of optimization

largest slice of pizza = maximum area of the sector

Step 2: Formula for area and perimeter of sector:

Area of sector, A = 1/2r²θ

Perimeter of sector, P = 2r + rθ

Step 3: Determining the constraint

The perimeter, P, must be equal to 36 inches ie. 2r + rθ = 32

Therefore, θ = 32 -2r/r

Substituting θ in the formula for area;

A = 1/2r²(32 -2r/r)

A = -2r² + 16r

The domain of A(r) = 0 < r < 16

Step 4: Determine the maximum of object function

Critical point of A(r) , A'(r) = - 2r + 16

when A' = 0

r = -16/-2

r = 8

since, diameter, d = 2r,

Therefore, a diameter of 16 inches will give the largest slice