Problem 114 - Precalculus : Building Concepts and Connections

Question

The number of copies of a popular mystery writer's newest release sold at a local bookstore during each month after its release is given by \(n(x)=-5 x+100\) The price of the book during each month after its release is given by \(p(x)=-1.5 x+30 .\) Find \((n p)(3) .\) Interpret your results.

Step-by-Step Solution

Answer

The total revenue from the book in the third month after its release is $2,167.5.

Step 1: Identify the Functions

The functions given are \(n(x)=-5x+100\) for the number of copies sold each month, and \(p(x)=-1.5x+30\) for the price of the book each month.

Step 2: Evaluate n(3) and p(3)

Substitute \(x=3\) into both functions to find the number of copies sold and the price in the third month. Thus, \(n(3)=-5(3)+100=85\) and \(p(3)=-1.5(3)+30=25.5\). After three months, 85 copies are sold at a price of $25.5 per copy.

Step 3: Evaluate (n p)(3)

Multiply the values of \(n(3)\) and \(p(3)\) to find the total revenue from the book in the third month. \((n p)(3) = n(3) * p(3) = 85 * 25.5 = $2,167.5\). So, the total revenue from the book in the third month is $2,167.5.

Key Concepts

Algebra

Algebra is a foundational branch of mathematics that involves symbols and the rules for manipulating these symbols to solve equations and model real-world scenarios. In the given exercise, we deal with linear equations to understand and predict the business performance of book sales over time. With algebra, you can express real-world problems using mathematical expressions. Here, two algebraic functions represent relationships over time:

• The number of books sold each month: \( n(x) = -5x + 100 \)
• The price per book each month: \( p(x) = -1.5x + 30 \)

Each equation is linear, where their terms can be plotted on a graph as straight lines. The coefficients ((-5) for \(n(x)\) and (-1.5) for \(p(x)\)) show how values change each month. As time (x) increases, the number of books sold decreases while the book price also decreases. These linear relationships model a typical behavior in business contexts, where demand might decrease over time while prices adjust in response.

Function Evaluation

Function evaluation involves plugging specific input values into a function to find the corresponding output. It is an essential skill when working with equations to gain precise insights into different scenarios. In our exercise, we evaluate the functions \(n(x)\) and \(p(x)\) for \(x = 3\) because we want to understand the sales and price situation of the book in the third month:

• For the book sales: \(n(3) = -5(3) + 100 = 85\). This means that 85 books were sold.
• For the book price: \(p(3) = -1.5(3) + 30 = 25.5\). This shows the price of each book was $25.50.

Being able to perform function evaluation allows us to directly determine numerical results from analytical expressions. This practice builds a crucial bridge between abstract formulas and practical real-world applications, making it indispensable for thorough analysis.

Revenue Calculation

Revenue calculation involves determining the total income from sales of goods or services. It's a core concept in business and economics, crucial for understanding and optimizing financial performance. In this exercise, we calculate revenue for the third month by evaluating the product of two functions \(n(x)\) and \(p(x)\):

• Total Revenue = \((n \cdot p)(3) = n(3) \times p(3) = 85 \times 25.5 = 2167.5\).

This calculation shows that, after three months, the total revenue from the book sales was $2,167.50. Each month's revenue reveals how pricing and sales numbers collaborate to impact overall earnings. By multiplying the outputs of our functions \(n(x)\) and \(p(x)\), we can derive a powerful measure of economic output, allowing businesses to strategize for improved profitability and market presence.