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!