Suppose a company has fixed costs of $800 and variable costs per unit of 
7
8
x + 1240 dollars,
 where x is the total number of units produced. Suppose further that the selling price of its product is 
1300 − 
1
8
x dollars per unit.
(a) Find the break-even points. (Enter your answers as a comma-separated list.)
x = 


To find the break-even points, we need to determine when the company's revenue equals its total cost.

### Step 1: Define the equations for total cost and total revenue.

- **Total cost** consists of fixed costs plus variable costs:
  \[
  \text{Total Cost} = \text{Fixed Cost} + \text{Variable Cost} = 800 + \left(\frac{7}{8}x + 1240\right)
  \]
  Simplifying the total cost:
  \[
  \text{Total Cost} = 800 + \frac{7}{8}x + 1240 = \frac{7}{8}x + 2040
  \]

- **Revenue** is the price per unit multiplied by the number of units sold:
  \[
  \text{Revenue} = \left(1300 - \frac{1}{8}x\right) \times x
  \]
  Expanding this expression:
  \[
  \text{Revenue} = 1300x - \frac{1}{8}x^2
  \]

### Step 2: Set the total revenue equal to the total cost to find the break-even points.
\[
1300x - \frac{1}{8}x^2 = \frac{7}{8}x + 2040
\]

### Step 3: Rearrange the equation and simplify.
Move all terms to one side of the equation:
\[
1300x - \frac{1}{8}x^2 - \frac{7}{8}x - 2040 = 0
\]
Combine like terms:
\[
-\frac{1}{8}x^2 + \left(1300 - \frac{7}{8}\right)x - 2040 = 0
\]
Simplifying further:
\[
-\frac{1}{8}x^2 + \frac{10393}{8}x - 2040 = 0
\]
Multiply through by 8 to eliminate fractions:
\[
-x^2 + 10393x - 16320 = 0
\]

### Step 4: Solve the quadratic equation.
The equation is:
\[
x^2 - 10393x + 16320 = 0
\]

Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -10393$, and $c = -16320$, let's compute the solutions.

I’ll calculate the values for you.The solutions to the quadratic equation are:

\[
x_1 \approx 10394.57 \quad \text{and} \quad x_2 \approx -1.57
\]

Since the number of units produced cannot be negative, we discard the negative solution. Thus, the break-even point is approximately:

\[
x \approx 10394.57
\]

Would you like more details on any step or have further questions?

Here are 5 related questions:
1. How do you interpret the break-even point in terms of profit?
2. What happens if fixed costs increase? How does it affect the break-even point?
3. How can a company reduce its break-even point?
4. What does a break-even analysis assume about variable costs and revenue per unit?
5. Can a company have multiple break-even points? If so, under what conditions?

**Tip**: Always compare fixed and variable costs when analyzing profitability; a high fixed cost requires a higher volume to break even.

Suppose a company has fixed costs of $800 and variable costs per unit of 7/8x + 1240 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is 1300 - 1/8x dollars per unit. (a) Find the break-even points. (Enter your answers as a comma-separated list.) x = 

Discover how to find the break-even points for a company's production costs using a quadratic equation. This example involves solving for x in a real-world break-even analysis problem, including fixed and variable costs. Suitable for high school students in Grades 10-12.

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