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A furniture company is introducing a new line of lounge chairs next quarter. These are the cost and revenue functions, where [tex]$x$[/tex] represents the number of chairs to be manufactured and sold:

[tex]\[
\begin{array}{l}
R(x) = 1,248x - 8.32x^2 \\
C(x) = 36,400 - 83.2x
\end{array}
\][/tex]

For the company to make a profit on the chairs, the selling price can go no lower than [tex]$\boxed{\phantom{0}}$[/tex] and no higher than [tex]$\boxed{\phantom{0}}$[/tex].



Answer :

GinnyAnswer
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.

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