Question

Suppose you are offered just one slice from a round pizza (in other words, a sector of a circle). The slice must have a perimeter of 32 inches. What diameter pizza will reward you with the largest slice? Specify the diameter and the maximum area of the slice.

Answer 1

The answer is diameter: 32 in, area: 64 in²

The perimeter of a sector of a circle is:
P = 2r + l
l = rθ
P = 2r + rθ

P = 32 in
32 = 2r + rθ
32 - 2r = rθ
θ = (32 - 2r)/r
θ = (2*16 - 2*r)/r
θ = 2(16 - r)/r

Area of the sector of the circle is:
A = r²/2 * θ
A = r²/2 * 2(16 - r)/r
A = r² * (16 - r)/r
A = r(16 - r)
A = 16r - r²

For the maximum area:
A' = 16 - 2r
A' = 0
16 - 2r = 0
16 = 2r
r = 8 in

The diameter (D) of the circle is twice of the radius:
D = 2r = 2 * 8 = 16 in

The maximum area is:
A = 16r - r²
r = 8 in
A = 16 * 8 - 8²
A = 128 - 64
A = 64 in²