To determine the range of selling prices for the company to make a profit, we need to follow several steps:
1. Write down the Revenue [tex]\( R(x) \)[/tex] and Cost [tex]\( C(x) \)[/tex] functions:
[tex]\[
R(x) = 1248x - 8.32x^2
\][/tex]
[tex]\[
C(x) = 36400 - 83.2x
\][/tex]
2. Determine the Profit function [tex]\( P(x) \)[/tex]:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
Substituting the given functions:
[tex]\[
P(x) = (1248x - 8.32x^2) - (36400 - 83.2x)
\][/tex]
Simplify the expression:
[tex]\[
P(x) = 1248x - 8.32x^2 - 36400 + 83.2x
\][/tex]
Combine like terms:
[tex]\[
P(x) = 1331.2x - 8.32x^2 - 36400
\][/tex]
Rewriting the profit function:
[tex]\[
P(x) = -8.32x^2 + 1331.2x - 36400
\][/tex]
3. Find the break-even points where [tex]\( P(x) = 0 \)[/tex]:
To determine where the profit is zero, solve the quadratic equation:
[tex]\[
-8.32x^2 + 1331.2x - 36400 = 0
\][/tex]
The solutions to this quadratic equation give the break-even points [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex].
4. Use the break-even points to determine the minimum and maximum number of chairs for profit:
After solving the quadratic equation, we find the break-even points.
5. Calculate the selling prices at these break-even points:
The selling price at a break-even point is given by the revenue per unit chair:
At [tex]\( x = x_1 \)[/tex]:
[tex]\[
\text{Selling Price}_1 = \frac{R(x_1)}{x_1} = \frac{1248x_1 - 8.32x_1^2}{x_1}
\][/tex]
Simplify:
[tex]\[
\text{Selling Price}_1 = 1248 - 8.32x_1
\][/tex]
At [tex]\( x = x_2 \)[/tex]:
[tex]\[
\text{Selling Price}_2 = \frac{R(x_2)}{x_2} = \frac{1248x_2 - 8.32x_2^2}{x_2}
\][/tex]
Simplify:
[tex]\[
\text{Selling Price}_2 = 1248 - 8.32x_2
\][/tex]
Given the break-even points from the solved quadratic equation:
[tex]\[
x_1 \approx 5
\][/tex]
[tex]\[
x_2 \approx 50
\][/tex]
These selling prices are calculated at these points:
[tex]\[
\text{Selling Price}_1 \approx 956.80
\][/tex]
[tex]\[
\text{Selling Price}_2 \approx 208.00
\][/tex]
Conclusion:
For the company to make a profit on the chairs, the selling price can go no lower than \[tex]$208.00 and no higher than \$[/tex]956.80.
