Answer
The diameter should be $~~16~in$
Work Step by Step
Let $\theta$ be the central angle of the pizza slice.
Let $r$ be the radius of the pizza.
We can use the perimeter to find an expression for $\theta$:
$r+r+\theta~r = 32$
$\theta = \frac{32-2r}{r}$
We can find an expression for the area of the pizza slice:
$A = (\frac{\theta}{2\pi})~\pi~r^2$
$A = \frac{\theta~r^2}{2}$
$A = \frac{(32-2r)~r^2}{2r}$
$A = (16-r)~r$
$A = 16r-r^2$
We can find $\frac{dA}{dr}$:
$\frac{dA}{dr} = 16-2r = 0$
$r = 8$
Since the diameter is twice the radius, the diameter should be $~~16~in$
