Question

The function f(x) = -(x-20)(x - 100) represents a company's monthly profit as a function of x, the number of purchase

orders received. Which number of purchase orders will generate the greatest profit?
20
60
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Answer 1

The number of purchase orders that will generate the greatest profit is 60 orders

Explanation:

we have

`f(x)=-(x-20)(x-100)`

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

`f(x)=-(x-20)(x-100)`

`f(x)=-(x^2-100x-20x+2,000)`

`f(x)=-(x^2-120x+2,000)`

Complete the square

`f(x)=-(x^2-120x)-2,000`

`f(x)=-(x^2-120x+3,600)-2,000+3,600`

`f(x)=-(x^2-120x+3,600)+1,600`

Rewrite as perfect squares

`f(x)=-(x-60)^2+1,600`

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