Question

Falcon manufacturing estimates that its weekly profit p in hundred of dollars can be approximated by the formula below where x is the number of units produced per week in thousands How many units should the company produce per week to earn the maximum profit ?find the maximum weekly profit

Answer 1

The Quadratic Function given in the exercise is:

`P=-3x^2+6x+10`

Where "P" is the weekly profit of Dalco Manufacturing, in hundreds of dollars. And "x" is the number of units produced per week, in thousands.

By definition, when you graph a Quadratic Function, you obtain a parabola.

The vertex of a parabola is its maximum point when the Leading Coefficient of the function is negative. In this case, you can identify that the Leading Coefficient is:

`a=-3`

Therefore, the maximum point of the parabola is its vertex.

Knowing the explained above, you can determine that if you find the vertex of this parabola, you will be able to solve the exercise.

1. You need to find the x-coordinate of the vertex. You can use this formula:

`x=(-b)/(2a)`

In this case:

`\begin{gathered} a=-3 \\ b=6 \end{gathered}`

Therefore, you get that:

`x=(-6)/((2)(-3))=(-6)/(-6)=1`

2. Finally, to find the y-coordinate of the vertex, you need to substitute that value of "x" into the original equation and evaluate:

`\begin{gathered} P=-3(1)^2+6(1)+10=-3+6+10=13 \\ \end{gathered}`

3. Then, the vertex is:

`(1,13)`

According to the information given in the exercise, "x" is the number of units produced per week, in thousands. This means that actually, the x-coordinate of the vertex in thousands is:

`x=1,000`

Therefore, the answer is: The company should produce 1,000 units per week to earn the maximum profit.