Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

A certain paperback sells for \(\$ 12\). The author is paid royalties of \(10 \%\) on the first 10,000 copies sold, \(12.5 \%\) on the next 5000 copies, and \(15 \%\) on any additional copies. Find a piecewise-defined function \(R\) that specifies the total royalties if \(x\) copies are sold.

Step-by-Step Solution

Answer

The piecewise function is: \\ \(R(x) = 1.2x\) for \(0 \leq x \leq 10,000;\)\\ \(R(x) = 12,000 + 1.5(x - 10,000)\) for \(10,000 < x \leq 15,000;\)\\ \(R(x) = 19,500 + 1.8(x - 15,000)\) for \(x > 15,000\).

Step 1: Understand the Royalties Tiers

The royalties are calculated in three tiers: 1. 10% for the first 10,000 copies. 2. 12.5% for the next 5,000 copies, making up to a total of 15,000 copies. 3. 15% for any copies beyond 15,000.

Step 2: Calculate Royalties for the First Tier

For the first 10,000 copies sold, the royalties, if sold, are calculated as 10% of the sales price:\[ R(x) = 0.10 \times 12 \times x \text{ for } 0 \leq x \leq 10,000 \]

Step 3: Calculate Royalties for the Second Tier

If sales exceed 10,000 copies but do not exceed 15,000 copies, calculate royalties as follows:For the first 10,000 copies:\[ R_1 = 0.10 \times 12 \times 10,000 \]For the next \(x - 10,000\) copies:\[ R_2 = 0.125 \times 12 \times (x - 10,000) \]So, for \(10,000 < x \leq 15,000\):\[ R(x) = R_1 + R_2 = 12,000 + 1.5(x - 10,000) \]

Step 4: Calculate Royalties for the Third Tier

For sales exceeding 15,000 copies, the royalties include:First 10,000 copies:\[ R_1 = 0.10 \times 12 \times 10,000 = 12,000 \]Next 5,000 copies:\[ R_2 = 0.125 \times 12 \times 5,000 = 7,500 \]And for any additional \(x - 15,000\) copies:\[ R_3 = 0.15 \times 12 \times (x - 15,000) \]Thus, for \(x > 15,000\):\[ R(x) = 12,000 + 7,500 + 1.8(x - 15,000) \]

Step 5: Define the Piecewise Function

Combine the results from previous steps to define the piecewise function:For \(0 \leq x \leq 10,000\):\[ R(x) = 1.2x \]For \(10,000 < x \leq 15,000\):\[ R(x) = 12,000 + 1.5(x - 10,000) \]For \(x > 15,000\):\[ R(x) = 19,500 + 1.8(x - 15,000) \]

Key Concepts

Royalties Calculation

Royalties are payments made to authors or creators based on the sales of their products, typically as a percentage of the sale prices. In this exercise, royalties are calculated in stages, depending on the number of books sold.
To understand how to calculate these royalties, let's break down each stage or tier:

• Tier 1: For the first 10,000 books sold, the author gets 10% of the sales. At a book price of $12, this translates to a royalty of \(0.10 \times 12 \times x\), where \(x\) is the number of copies within this tier (0 to 10,000).
• Tier 2: For the next 5,000 copies (from 10,001 to 15,000), the royalty rate increases to 12.5%. So, the royalty for these books is \(0.125 \times 12 \times (x - 10,000)\). This amount is added to the first tier royalties for total sales within 15,000 copies.
• Tier 3: After 15,000 copies sold, the royalty rate jumps to 15%. The calculation for these copies is \(0.15 \times 12 \times (x - 15,000)\), added to the royalties from the first 15,000 copies.

This piecewise function effectively models different rates of royalties paid out depending on book sale intervals.

Tiered Pricing

Tiered pricing refers to a pricing strategy where the price per unit changes based on the quantity purchased or sold. In this exercise, the 'price' per copy refers to the rate of royalties the author earns.
This concept mirrors how businesses use tiered pricing to incentivize bulk purchases or recognize loyalty. The author here earns more per book as more copies are sold:

• Starting Tier: A base royalty rate of 10% can encourage initial sales volume with a reasonable perk for the author.
• Middle Tier: An increase to 12.5% rewards reaching higher sales, reflecting an incentive for driving additional book promo efforts.
• Premium Tier: At a sale count beyond 15,000, a 15% royalty not only handsomely compensates the author but also pushes for sustaining the market momentum.

The implementation of a tiered pricing strategy allows shareholders like authors and publishers to align monetary rewards with sales metrics.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world scenarios. In this example involving book sales and royalties, mathematical modeling comes to life through piecewise functions.
With piecewise functions, we can succinctly express complex systems that involve different conditions or segments, as in the royalty payout example:

• First Segment: The function \(R(x) = 1.2x\) describes the region where sales are between 0 and 10,000 copies.
• Second Segment: For sales between 10,001 to 15,000, it's efficiently represented as \(R(x) = 12,000 + 1.5(x - 10,000)\).
• Third Segment: Sales exceeding 15,000 adopts the formula \(R(x) = 19,500 + 1.8(x - 15,000)\).

This mathematical structuring allows us to not only calculate outcomes accurately but also visually understand and adjust to different business needs or conditions. By breaking down the problem into logical parts, complex situations like tiered royalties calculation can be simplified and effectively managed.