Answer
Diameter $16$ $in.$
Work Step by Step
The perimeter of a slice is
$2r+r\theta=32$
so we have:
$\theta=\frac{32-2r}{r}$.
The area $A$ of the slice is
$A$ = $\frac{1}{2}{r^{2}}{θ}$ = $\frac{1}{2}{r^{2}}\left(\frac{32-2r}{r}\right)$ = $16r-r^{2}$ for $0$ $\leq$ $r$ $\leq$ $16$
$A'(r)$ = $16-2r$
$A'(r)$ = $0$
$16-2r$ = $0$
$r$ = $8$
Since $A(0)$ = $0$, $A(16)$ = $0$ and $A(8)$ = $64$,
the largest piece comes from a pizza with radius $8$ $in.$ and diameter $16$ $in.$
$θ$ = $2$ radians $\approx$ $114.6°$ which is about $32$% of the whole pizza.