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 24 inches, what diameter pizza will reward you with the largest slice? in

A 12 in diameter will reward you with the largest slice of pizza.

Step-by-step explanation:

Let be the radius and be the angle of a circle.

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

The arc lenght of a circle with radius and angle is

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

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

Next, we substitute this equation into the area equation

The domain of the area is

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.

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

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

The second derivative is

Because the largest slice is when r = 6 in.

The diameter of the pizza is given by

A 12 in diameter will reward you with the largest slice of pizza.

thats not a system of equations, that is a solution