 Profit. The financial department for the company in

Problems 86 and 88 established the following cost function

for producing and selling x thousand notebook computers:

C1x2 = 4,000 + 500x thousand dollars

(A) Write a profit function for producing and selling x thou-

sand notebook computers and indicate its domain.

(B) Complete Table 13, computing profits to the nearest

thousand dollars.

Table 13 Profit

x (thousands) P1x2 (thousand $)

1 -2,560

5

10

15

20

25

(C) Plot the points in part (B) and sketch a graph of the

profit function using these points.

Let's break down the problem step by step:

### Part (A): Write a Profit Function and Indicate Its Domain

**1. Cost Function:**

Given the cost function for producing and selling $ x $ thousand notebook computers:
\[
C(x) = 4000 + 500x \text{ (in thousand dollars)}
\]

**2. Revenue Function:**

Assume the selling price per thousand units is $ p $ thousand dollars. The revenue function is:
\[
R(x) = p \times x \text{ (in thousand dollars)}
\]

**3. Profit Function:**

The profit function is defined as the difference between revenue and cost:
\[
P(x) = R(x) - C(x)
\]
\[
P(x) = p \times x - (4000 + 500x)
\]
\[
P(x) = (p - 500)x - 4000 \text{ (in thousand dollars)}
\]

**Domain of the Profit Function:**

Since $ x $ represents the number of thousands of notebook computers produced and sold, $ x $ must be a non-negative number. Therefore, the domain of $ P(x) $ is:
\[
\text{Domain: } x \geq 0
\]

### Part (B): Complete Table 13

To fill out Table 13, we need to compute the profit for the given values of $ x $. However, the selling price $ p $ is not provided, so we can't compute the exact profit values. If we assume $ p = 1000 $ (as a common price), we can continue.

Using $ p = 1000 $:
\[
P(x) = (1000 - 500)x - 4000
\]
\[
P(x) = 500x - 4000
\]

Now, calculate $ P(x) $ for each value of $ x $:

\[
P(1) = 500(1) - 4000 = 500 - 4000 = -3500 \text{ thousand dollars}
\]
\[
P(5) = 500(5) - 4000 = 2500 - 4000 = -1500 \text{ thousand dollars}
\]
\[
P(10) = 500(10) - 4000 = 5000 - 4000 = 1000 \text{ thousand dollars}
\]
\[
P(15) = 500(15) - 4000 = 7500 - 4000 = 3500 \text{ thousand dollars}
\]
\[
P(20) = 500(20) - 4000 = 10000 - 4000 = 6000 \text{ thousand dollars}
\]
\[
P(25) = 500(25) - 4000 = 12500 - 4000 = 8500 \text{ thousand dollars}
\]

### Part (C): Plot the Points and Sketch the Graph

Let's summarize the computed values in the table:

| $ x $ (thousands) | $ P(x) $ (thousand $) |
|:------------------:|:----------------------:|
| 1                  | -3500                  |
| 5                  | -1500                  |
| 10                 | 1000                   |
| 15                 | 3500                   |
| 20                 | 6000                   |
| 25                 | 8500                   |

Finally, the graph of the profit function $ P(x) = 500x - 4000 $ is a straight line, with the slope 500 and y-intercept -4000.

Would you like a detailed graph to visualize this, or any additional explanations? Here are 5 related questions to expand on this topic:

1. What would the profit function look like if the cost per unit increased?
2. How does changing the selling price affect the profit function?
3. What is the break-even point for this profit function?
4. How would the profit change if fixed costs decreased?
5. Can the profit function be nonlinear, and under what circumstances might this occur?

**Tip:** Remember, the slope of the profit function corresponds to the difference between the selling price per unit and the variable cost per unit.

The financial department for the company in Problems 86 and 88 established the following cost function for producing and selling x thousand notebook computers: C(x) = 4000 + 500x thousand dollars. (A) Write a profit function for producing and selling x thousand notebook computers and indicate its domain. (B) Complete Table 13, computing profits to the nearest thousand dollars. (C) Plot the points in part (B) and sketch a graph of the profit function using these points.

Explore the profit function derived from the cost function for producing notebook computers. Learn how to write, analyze, and graph profit functions using real data points. Suitable for students in Grades 10-12 studying algebra and linear functions.

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