Profit The monthly profit P for a widget producer is a function of the number n of widgets sold. The formula is
P= -15 plus 10n minus 0.2n exponent 2
Here P is measured in thousands of dollars, n is measured in thousands of widgets, and the formula is valid up to a level of 15 thousand widgets sold.

a. Make a graph of P versus n.

b. Calculate P(1) and explain in practical terms
what your answer means.

c. Is the graph concave up or concave down?
Explain in practical terms what this means.

d. The break-even point is the sales level at which the profit is 0. Find the break-even point for this widget producer.
Solve a only

Here is the graph of profit \( P \) (in thousands of dollars) versus the number of widgets sold \( n \) (in thousands). The curve represents the profit function \( P(n) = -15 + 10n - 0.2n^2 \). 

This graph shows how profit changes as the number of widgets sold increases. Would you like to continue with part (b) or have any questions about the graph?

Here are 5 related questions:
1. What does the shape of the graph tell you about maximum profit?
2. How would the graph change if the production costs decreased?
3. How do real-world factors like demand affect profit functions?
4. What is the significance of negative profits in a business scenario?
5. What happens to the graph as \( n \) approaches its limits?

**Tip**: To find optimal production levels, look at the vertex of the parabola in the profit graph.

The monthly profit P for a widget producer is a function of the number n of widgets sold. The formula is P = -15 + 10n - 0.2n^2. P is measured in thousands of dollars, n is measured in thousands of widgets, and the formula is valid up to a level of 15 thousand widgets sold. Solve part (a): Make a graph of P versus n.

Explore how to graph the profit function P(n) = -15 + 10n - 0.2n^2, representing the profit in thousands of dollars based on the number of widgets sold. This problem demonstrates quadratic functions and is suitable for students in Grades 9-12.

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