Your college newspaper, The Collegiate Investigator, has fixed production costs of $74 per edition and marginal printing and distribution costs of 41¢ per copy. The Collegiate Investigator sells for 51¢ per copy.
(a)
Write down the associated cost function C(x) in dollars.
C(x) =

Write down the revenue function R(x) in dollars.
R(x) =

Write down the profit function P(x) in dollars.
P(x) =

(b)
What profit (or loss) results from the sale of 500 copies of The Collegiate Investigator?
$
(c)
How many copies should be sold in order to break even?
copies

Let's break down the given problem step by step.

### (a) Cost, Revenue, and Profit Functions

#### 1. **Cost Function $ C(x) $:**
- The fixed cost is $74 per edition.
- The marginal (per copy) cost is 41¢ (which is $0.41) per copy.
Thus, the total cost function $ C(x) $ in dollars (where $ x $ is the number of copies) is:
\[
C(x) = 74 + 0.41x
\]

#### 2. **Revenue Function $ R(x) $:**
- The selling price per copy is 51¢ (which is $0.51).
Thus, the revenue function $ R(x) $ in dollars is:
\[
R(x) = 0.51x
\]

#### 3. **Profit Function $ P(x) $:**
- Profit is the difference between revenue and cost.
Thus, the profit function $ P(x) $ is:
\[
P(x) = R(x) - C(x) = 0.51x - (74 + 0.41x)
\]
Simplifying:
\[
P(x) = 0.51x - 74 - 0.41x = 0.10x - 74
\]

### (b) Profit (or Loss) from the Sale of 500 Copies

We can calculate the profit when $ x = 500 $ copies are sold by substituting $ x = 500 $ into the profit function:
\[
P(500) = 0.10(500) - 74 = 50 - 74 = -24
\]
Thus, the newspaper incurs a loss of $24 when 500 copies are sold.

### (c) Break-even Point

At the break-even point, profit equals zero. So, we set the profit function $ P(x) = 0 $ and solve for $ x $:
\[
0.10x - 74 = 0
\]
\[
0.10x = 74
\]
\[
x = \frac{74}{0.10} = 740
\]
Thus, 740 copies need to be sold to break even.

---

### Summary of answers:
**(a)**  
Cost function: $ C(x) = 74 + 0.41x $  
Revenue function: $ R(x) = 0.51x $  
Profit function: $ P(x) = 0.10x - 74 $

**(b)** Profit (or loss) from 500 copies: $-24 (loss)

**(c)** Break-even point: 740 copies

---

Would you like a more detailed explanation on any part of this? Here are some related questions:

1. What happens to profit if the price per copy changes?
2. How would fixed costs affect the break-even point?
3. How does an increase in sales volume impact overall profit?
4. What if the marginal cost per copy increased to 45¢?
5. How is break-even analysis useful for businesses?

**Tip:** When calculating profits, always ensure to subtract all costs, including both fixed and variable, from your total revenue!

Your college newspaper, The Collegiate Investigator, has fixed production costs of $74 per edition and marginal printing and distribution costs of 41¢ per copy. The Collegiate Investigator sells for 51¢ per copy. (a) Write down the associated cost function C(x) in dollars. C(x) = Write down the revenue function R(x) in dollars. R(x) = Write down the profit function P(x) in dollars. P(x) = (b) What profit (or loss) results from the sale of 500 copies of The Collegiate Investigator? $ (c) How many copies should be sold in order to break even? copies

Learn how to determine the cost, revenue, and profit functions for a college newspaper, The Collegiate Investigator. This problem involves analyzing fixed and marginal costs, calculating profit or loss for 500 copies, and determining the break-even point. Suitable for students in Grades 9-12.

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