https://file.aiquickdraw.com/mathbot/file/1727313929219xqk66ggx.jpg

### (D) Break-even points and graph:
- The break-even points, where \( P(x) = 0 \), occur at:
  \[
  x = 280 \quad \text{and} \quad x = 1520
  \]

These points represent the values of \( x \) (number of ski jackets) where the profit is zero, meaning the revenue equals the cost.

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### Related questions:
1. How do we interpret the horizontal tangents in the context of business?
2. What is the significance of the break-even points in terms of profitability?
3. How would the profit function change if the fixed costs in \( C(x) \) were different?
4. What is the maximum profit, and how can we determine it from the profit function?
5. How do changes in the cost function affect the overall break-even points?

### Tip:
Remember, horizontal tangents occur when the rate of change (slope) of a function is zero, indicating a local maximum or minimum for revenue or profit.

The total cost and the total revenue (in dollars) for the production and sale of x ski jackets are given by C(x) = 20x + 42,560 and R(x) = 200x - 0.1x^2 for 0 ≤ x ≤ 2000. Find the value of x where the graph of R(x) has a horizontal tangent line. Find the profit function P(x). Find the value of x where the graph of P(x) has a horizontal tangent line. Graph C(x), R(x), and P(x) on the same coordinate system for 0 ≤ x ≤ 2000. Find the break-even points. Find the x-intercepts of the graph of P(x).

Analyze the revenue, cost, and profit functions for ski jackets using calculus. Learn how to find break-even points, horizontal tangents, and graph functions. Suitable for Grade 12 and above, this problem covers essential business applications of calculus.

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