Question

If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 32 inches, what diameter pizza will reward you with the largest slice

Answer 1

A 16 inches diameter will reward you with the largest slice of pizza.

Explanation:

Let r be the radius and
`\theta` be the angle of a circle.

According with the graph, the area of the sector is given by

`A=(1)/(2)r^2\theta`

The arc length of a circle with radius r and angle
`\theta` is r
`\theta`

The perimeter of the pizza slice is composed of two straight pieces, each of length r inches, and an arc of the circle which you know has length s = rθ inches. Thus the perimeter has length

The perimeter of the pizza slice is composed of two straight pieces, each of length r inches, and an arc of the circle which you know has length s = rθ inches.

Thus the perimeter has length

`2r+r\theta=32 \:in`

We need to express the area as a function of one variable, to do this we use the above equation and we solve for
`\theta`

`2r+r\theta=32\\\\r\theta=32-2r\\\\\theta=(32-2r)/(r)`

Next, we substitute this equation into the area equation

`A=(1)/(2)r^2((32-2r)/(r))\\\\A=(1)/(2)r(32-2r)\\\\A=16r-r^2`

The domain of the area is

`0<r<12`

To find the diameter of pizza that will reward you with the largest slice you need to find the derivative of the area and set it equal to zero to find the critical points.

`(d)/(dr) A=(d)/(dr)(16r-r^2)\\\\A'(r)=(d)/(dr)(16r)-(d)/(dr)(r^2)\\\\A'(r)=16-2r16-2r=0\\\\-2r=-16\\\\(-2r)/(-2)=(-16)/(-2)\\\\r=8`

To check if r=8 is a maximum we use the Second Derivative test

if
`f'(c)=0` and
`f''(c)<0` , then f(x) has a local maximum at x = c.

The second derivative is

`(d)/(dr) A'(r)=(d)/(dr) (16-2r)\\\\A''(r)=-2`

Because
`A''(r)=-2 <0` the largest slice is when r = 8 in.

The diameter of the pizza is given by

`D=2r=2\cdot 8=16 \:in`

A 16 inches diameter will reward you with the largest slice of pizza.