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.