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Let's break down the problem and solve it step by step. The information provided is as follows:

- Fixed production costs: \$72 per edition
- Marginal printing and distribution costs: 39¢ (or \$0.39) per copy
- Sale price of the newspaper: 49¢ (or \$0.49) per copy

### (a) Write down the functions:

1. **Cost function** $ C(x) $ (in dollars):
   - The fixed cost is \$72, and the marginal cost per copy is \$0.39. Therefore, the cost function is:
   \[
   C(x) = 72 + 0.39x
   \]
   where $ x $ is the number of copies printed.

2. **Revenue function** $ R(x) $ (in dollars):
   - The revenue from selling each copy is \$0.49. Therefore, the revenue function is:
   \[
   R(x) = 0.49x
   \]

3. **Profit function** $ P(x) $ (in dollars):
   - Profit is the difference between revenue and cost. Thus, the profit function is:
   \[
   P(x) = R(x) - C(x) = 0.49x - (72 + 0.39x)
   \]
   Simplifying the profit function:
   \[
   P(x) = 0.49x - 72 - 0.39x = 0.1x - 72
   \]

### (b) Profit or loss for 500 copies:

Substitute $ x = 500 $ into the profit function:
\[
P(500) = 0.1(500) - 72 = 50 - 72 = -22
\]
So, the profit (or loss) from selling 500 copies is **\$-22**, meaning a loss of \$22.

### (c) Break-even point:

To find the break-even point, set the profit function $ P(x) = 0 $ and solve for $ x $:
\[
0 = 0.1x - 72
\]
\[
0.1x = 72
\]
\[
x = \frac{72}{0.1} = 720
\]
Therefore, **720 copies** must be sold to break even.

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Would you like more details on any step, or do you have any specific questions? Here are 5 related questions to explore further:

1. How does the fixed cost affect the total cost function as more copies are produced?
2. How would the profit function change if the selling price were increased?
3. What happens to the break-even point if the marginal cost per copy rises to \$0.45?
4. Can you explain the relationship between marginal cost and revenue in determining profit?
5. How would the analysis change if the fixed cost were lowered?

**Tip:** Always check how small changes in cost or revenue impact the break-even point for better strategic decision-making.

Your college newspaper, The Collegiate Investigator, has fixed production costs of $72 per edition and marginal printing and distribution costs of 39¢ per copy. The Collegiate Investigator sells for 49¢ per copy. Write down the cost, revenue, and profit functions. Calculate profit or loss from 500 copies and find the break-even point.

Learn how to calculate cost, revenue, and profit functions for The Collegiate Investigator newspaper, including a breakdown of the profit or loss from selling 500 copies and the calculation of the break-even point. This problem applies algebraic concepts to real-world business scenarios, suitable for students in Grades 9-12.

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