Suppose a company has fixed costs of $4000 and variable costs per unit of

15

16

*x* + 1270 dollars,

where *x* is the total number of units produced. Suppose further that the selling price of its product is

1400 −

1

16

*x* dollars per unit.

Find the break-even points. (Enter your answers as a comma-separated list.)

To find the break-even points, we need to set the total costs equal to the total revenue.

### Given:
- Fixed Costs (FC) = $4000
- Variable Cost per unit (VC) = $ 15 \times \frac{x}{16} + 1270 $
- Selling Price per unit (SP) = $ 1400 - \frac{1}{16} x $

### Total Cost (TC):
The total cost is the sum of fixed and variable costs:
\[
TC = FC + (VC \times x) = 4000 + \left(15 \times \frac{x}{16} + 1270\right) \times x
\]

### Total Revenue (TR):
Total revenue is given by:
\[
TR = SP \times x = \left(1400 - \frac{1}{16} x\right) \times x
\]

### Setting TC equal to TR:
\[
4000 + \left(15 \times \frac{x}{16} + 1270\right) \times x = \left(1400 - \frac{1}{16} x\right) \times x
\]

### Expanding both sides:
1. **Left Side:**
\[
TC = 4000 + \left(\frac{15x}{16} + 1270\right) x = 4000 + \frac{15x^2}{16} + 1270x
\]

2. **Right Side:**
\[
TR = \left(1400 - \frac{1}{16} x\right) x = 1400x - \frac{x^2}{16}
\]

### Combining:
\[
4000 + \frac{15x^2}{16} + 1270x = 1400x - \frac{x^2}{16}
\]

### Rearranging:
Combine all terms on one side:
\[
\frac{15x^2}{16} + \frac{x^2}{16} + 1270x - 1400x + 4000 = 0
\]
This simplifies to:
\[
\frac{16x^2}{16} - 130x + 4000 = 0
\]
\[
x^2 - 130x + 4000 = 0
\]

### Solving the Quadratic Equation:
Using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where $ a = 1 $, $ b = -130 $, and $ c = 4000 $.

Calculating the discriminant:
\[
b^2 - 4ac = (-130)^2 - 4 \times 1 \times 4000 = 16900 - 16000 = 900
\]

Calculating the roots:
\[
x = \frac{130 \pm \sqrt{900}}{2}
\]
\[
x = \frac{130 \pm 30}{2}
\]
This gives us two solutions:
\[
x = \frac{160}{2} = 80 \quad \text{and} \quad x = \frac{100}{2} = 50
\]

### Break-even Points:
The break-even points are:
\[
\boxed{50, 80}
\]

If you have any questions or need further details, feel free to ask! Here are some follow-up questions to consider:

1. How would changes in fixed costs affect the break-even points?
2. What impact do variable costs have on the overall profitability?
3. How can the company increase its break-even sales?
4. What strategies can be implemented if the selling price changes?
5. How would a decrease in selling price affect the break-even analysis? 

Tip: Always analyze how fixed and variable costs interact when assessing profitability!

Suppose a company has fixed costs of $4000 and variable costs per unit of 15/16*x + 1270 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is 1400 - 1/16*x dollars per unit. Find the break-even points.

Explore the process of finding break-even points for a company with fixed costs of $4000 and variable costs defined by a quadratic equation. Learn the key steps to set total costs equal to total revenue. Suitable for high school students.

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