To determine the selling price range for the company to make a profit, we need to evaluate where the revenue function [tex]\(R(x)\)[/tex] exceeds the cost function [tex]\(C(x)\)[/tex].
Given:
- Revenue function [tex]\(R(x) = 1248x - 8.32x^2\)[/tex]
- Cost function [tex]\(C(x) = 36400 - 83.2x\)[/tex]
The profit function [tex]\(P(x) = R(x) - C(x)\)[/tex] must be greater than zero for the company to make a profit.
Setting [tex]\(P(x) = 0\)[/tex] and solving for [tex]\(x\)[/tex] will give us the break-even points where the revenue equals the cost.
Given the solution for these break-even points:
- [tex]\(x_1 = 35\)[/tex]
- [tex]\(x_2 = 125\)[/tex]
Thus, the number of chairs sold (denoted by [tex]\(x\)[/tex]) must be between 35 and 125 for the company to make a profit.
So, the correct answers to fill in are:
For the company to make a profit on the chairs, the selling price can go no lower than \[tex]$ 35 and no higher than \$[/tex] 125.
