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Let's break down the problem shown in the image. The profit \( P \), in thousand dollars, is given as a quadratic function of the number of items \( N \) produced in a year:

\[
P(N) = -0.2N^2 + 3.6N - 9
\]

The questions are as follows:

### Part (a):
**Express using functional notation the profit at a production level of 2 items per year and calculate that value.**

We need to find \( P(2) \), i.e., substitute \( N = 2 \) into the given equation for \( P(N) \).

\[
P(2) = -0.2(2)^2 + 3.6(2) - 9
\]

I'll calculate this now.

### Part (b):
**Determine the two break-even points for this manufacturer.**  
To find the break-even points, set \( P(N) = 0 \) and solve the quadratic equation:

\[
-0.2N^2 + 3.6N - 9 = 0
\]

This is a quadratic equation, so we'll solve for \( N \) using the quadratic formula:

\[
N = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = -0.2 \), \( b = 3.6 \), and \( c = -9 \). I'll solve this as well.

### Part (c):
**Determine the maximum profit if the manufacturer can produce at most 21 items in a year.**

For a quadratic function \( P(N) = -0.2N^2 + 3.6N - 9 \), the maximum profit occurs at the vertex of the parabola, which can be found using the formula:

\[
N = -\frac{b}{2a}
\]

I'll calculate this value and then use it to find the corresponding maximum profit.

Let me go through the calculations for each part.Here are the results of the calculations:

### Part (a):
The profit at a production level of 2 items per year is:

\[
P(2) = -2.6 \text{ thousand dollars}
\]

### Part (b):
The two break-even points (where the profit is zero) are:

\[
N = 3 \quad \text{(smaller value)}, \quad N = 15 \quad \text{(larger value)}
\]

### Part (c):
The maximum profit occurs at a production level of \( N = 9 \) items per year, and the maximum profit is:

\[
P(9) = 7.2 \text{ thousand dollars}
\]

Would you like more details on any part of the calculations or further explanations? 

Here are five related questions to expand on this:
1. What is the significance of the break-even points in real-world business decisions?
2. How does changing the coefficients in a quadratic equation affect the shape of the graph?
3. What would happen to the maximum profit if the production limit was higher than 21 items?
4. How can we interpret a negative profit in part (a)?
5. How does the quadratic formula help in finding production values for different levels of profit?

**Tip**: Always ensure your equation is in the standard quadratic form \( ax^2 + bx + c = 0 \) before solving for the roots using the quadratic formula.

The profit P, in thousand dollars, that a manufacturer makes is a function of the number N of items produced in a year, given by P = -0.2N^2 + 3.6N - 9. Calculate the profit at a production level of 2, determine the break-even points, and find the maximum profit if the production is limited to 21 items per year.

Learn how to solve a quadratic profit function for a manufacturer, including calculating the profit at a specific production level, determining the break-even points, and finding the maximum profit for a production limit of 21 items. This problem covers key algebraic concepts such as the quadratic formula and vertex theorem, suitable for students in Grades 9-12.

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