Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

You ve written a book and have two publishers interested in putting it out. Both publishers anticipate selling the book for \(\$ 20 .\) The rst publisher guarantees you a at sum of \(\$ 8000\) for up to the rst 10,000 copies sold and will pay \(12 \%\) royalty for any copies sold in excess of 10,000 . For instance, if 10,001 copies were sold, you would receive \(\$ 8002.40 .\) The second publisher offers a royalty of \(10 \%\). Let \(x\) be the number of books sold. Let \(A(x)\) give the income under plan \(\mathrm{A}\) and \(B(x)\) give the income under plan \(\mathrm{B}\). (a) What is the algebraic formula for \(A(x)\) ? (b) What is the algebraic formula for \(B(x)\) ? (c) i. For what value(s) of \(x\) are the two plans equivalent? ii. For what values of \(x\) is plan B better? iii. For what values of \(x\) is plan \(\mathrm{A}\) better?

Step-by-Step Solution

Answer

The algebraic formula for A(x) is \(A(x) = 8000\) for \(x \leq 10000\) and \(A(x) = 8000 + 0.12 (x - 10000)\) for \(x > 10000\). The algebraic formula for B(x) is \(B(x) = 0.1 \times 20x\). To find the equivalent point we set A(x) equal to B(x) for \(x > 10000\), and solve the equation. For values of x less than the crossing point, plan A is better; for values of x greater than the crossing point, plan B is better.

Step 1: Formulate the algebraic formula (A(x)) for the first publisher

For the first publisher's plan A, the flat sum of \$8000 is given for the first 10000 copies sold. Any sold in excess of 10000 earns a 12\% royalty. So the formula for A(x) becomes: \[A(x) = 8000, \text{ when }x \leq 10000 \] \[A(x) = 8000 + 0.12(x - 10000), \text{ when } x > 10000 \]

Step 2: Formulate the algebraic formula (B(x)) for the second publisher

The second publisher offers a 10\% royalty for every book sold, thus the formula for B(x) is: \[B(x) = 0.1 \times 20x\] This gives the total income from x books sold as a 10\% royalty on a \$20 retail price per book.

Step 3: Determine the crossing point (x)

To find the value of x for which A(x) is equivalent to B(x), we set the two formulas equal to each other, remembering that this is only valid for x > 10000, due to the conditions of A(x). \[8000 + 0.12(x - 10000) = 0.1 \times 20x\] Solving this equation gives the crossing point of A(x) and B(x).

Step 4: Determine when each plan is better

For values of x less than the crossing point, plan A is better as it ensures an initial flat sum of \$8000 for the first 10000 copies sold. For values of x greater than the crossing point, plan B is better since the 10\% royalty per book will outweigh the benefits of the fixed sum and the 12\% royalty for extra copies in plan A.

Key Concepts

Algebraic formulas

Algebraic formulas are mathematical expressions used to represent relationships between variables. In this exercise, we deal with functions that relate the number of books sold to the revenue you earn under two different publisher plans.
For Plan A, the formula is split into two parts. If you sell up to 10,000 copies, you receive a fixed amount of \(8000. Beyond 10,000 copies, you earn \)8000 plus 12% of the revenue from additional copies:

• For up to 10,000 copies: \(A(x) = 8000\)
• For more than 10,000 copies: \(A(x) = 8000 + 0.12(x - 10000)\)

Plan B simplifies to a 10% royalty per book sold. If each book sells for $20, then:
\(B(x) = 0.1 \times 20x = 2x\)
This means your income from Plan B directly scales with the number of books sold.

Royalty calculations

Royalties are payments made to authors based on a percentage of sales. In this context, two royalty schemes are proposed by publishers.
Plan A guarantees a flat fee of \(8000 for up to 10,000 copies. For books sold beyond this threshold, they offer a 12% royalty. That means you earn additional income based on how much you sell after reaching 10,000 copies:
\[(x > 10000), \, \text{earn} \, 12\% \, \text{of} \, (\)20 \times \text{number of extra copies})\]
Plan B offers a 10% royalty on all sales. Therefore, you earn money proportional to each book sold without any initial fixed sum. This is beneficial if you expect high volumes since it scales linearly:
\[\text{Income} = 10\% \times 20 \times \text{total copies sold}\]
These different calculations can significantly impact your earnings based on the number of books sold.

Piecewise functions

Piecewise functions are those that have different expressions based on the input value. The revenue function for Plan A is a classic example of a piecewise function.
The Plan A revenue function breaks down into two segments:

• For \(x \leq 10000\): \(A(x) = 8000\). This part of the function reflects a constant revenue regardless of the exact number of books sold within this range.
• For \(x > 10000\): \(A(x) = 8000 + 0.12(x - 10000)\). Here, your revenue increases based on additional books sold beyond 10,000.

Such piecewise functions are useful for modeling scenarios with different phases or conditions. This approach provides flexibility for representing real-world situations where rules change at certain thresholds.
In contrast, flat-rate percentages, like Plan B, result in simpler linear functions that aren't piecewise. They're straightforward but may lack the adaptability required for variable sales thresholds.