Answer:
The number of purchase orders that will generate the greatest profit is 60 orders
Step-by-step explanation:
we have
[tex]f(x)=-(x-20)(x-100)[/tex]
This is a vertical parabola open downward
The vertex is a maximum
The y-coordinate of the vertex represent the greatest profit
The x-coordinate of the vertex represent the number of purchase orders for the greatest profit
Convert the function in vertex form
[tex]f(x)=-(x-20)(x-100)[/tex]
[tex]f(x)=-(x^2-100x-20x+2,000)[/tex]
[tex]f(x)=-(x^2-120x+2,000)[/tex]
Complete the square
[tex]f(x)=-(x^2-120x)-2,000[/tex]
[tex]f(x)=-(x^2-120x+3,600)-2,000+3,600[/tex]
[tex]f(x)=-(x^2-120x+3,600)+1,600[/tex]
Rewrite as perfect squares
[tex]f(x)=-(x-60)^2+1,600[/tex]
The vertex is the the point (60,1,600)
therefore
The greatest profit is $1,600
The number of purchase orders that will generate the greatest profit is 60 orders
