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.

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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

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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!