A small publishing company is releasing a new book. The production costs will include a one-time fixed cost for editing and an additional cost for each book printed. The total production cost \( C \) (in dollars) is given by the function \( \mathrm{C}=19.95 \mathrm{~N}+750 \), where \( N \) is the number of books. The total revenue earned (in dollars) from selling the books is given by the function \( R=34.60 \mathrm{~N} \). Let \( P \) be the profit made (In dollars). Write an equation relating \( P \) to N . Simplify your answer as much as possible. P=

First, let's define the profit \( P \) as the difference between total revenue \( R \) and total production cost \( C \). Using the given equations, we have: \[ P = R - C \] Substituting the expressions for \( R \) and \( C \): \[ P = 34.60N - (19.95N + 750) \] Now, simplify the equation: \[ P = 34.60N - 19.95N - 750 \] \[ P = (34.60 - 19.95)N - 750 \] \[ P = 14.65N - 750 \] So, the equation relating profit \( P \) to the number of books \( N \) is: \[ P = 14.65N - 750 \] --- In the world of publishing, small presses often face the challenge of balancing costs and revenue. A key to success is understanding the break-even point, which is the number of books that need to be sold to cover costs. For this situation, you can find the break-even point by setting \( P = 0 \) and solving for \( N \). This tells you how many copies need to fly off the shelves before profits start rolling in! From an entrepreneurial angle, maximizing profit isn't just about selling more books; it's also about managing costs effectively. Consider strategies like negotiating better printing prices or improving marketing efforts to boost sales. Small changes can have a large impact on that profit margin. Plus, connecting with readers via social media or local events can foster a community around your book, leading to even more sales!