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

remember that \[l=r \theta\]
and area of sector/slice is given by-\[Area=r^2 \frac{ \theta }{ 2 }\]

perimeter=2r+l
32=2r+rθ
(32-2r)/r =θ

put this theta in that area equation\[area=r^2 \frac{ \theta }{ 2 }\]
\[area=r^2 \frac{ (32-2r) }{ r }\]\[area=r(32-2r)\]\[area=32r-2r^2\]now we differentiate this equation with respect to r
doing this we will get the value of x for which area will be maximum
\[0=32-4r\]\[r=8\]
so when radius=8 we get the max area

we are asked the diameter

diameter=2x radius
diameter=2x8=16

we are also asked for area of slice
we got this-\[area=32r-2r^2\]
so just plug r=8
we get this-
area=128