Problem 45 - Precalculus: Mathematical for Calculus

Question

Publishing Books A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

Step-by-Step Solution

Answer

System of inequalities: \( f + n \leq 100,\ n \geq 20,\ f \geq n \).

Step 1: Identify Variables

Let's define variables for the types of books. Let \( f \) represent the number of fiction books published, and \( n \) represent the number of nonfiction books published each year.

Step 2: Express Total Books Constraint

The problem states that a total of no more than 100 books are published each year. This can be expressed as the inequality: \( f + n \leq 100 \).

Step 3: Nonfiction Minimum Constraint

The problem states that at least 20 books must be nonfiction. We express this as: \( n \geq 20 \).

Step 4: Fiction to Nonfiction Ratio

The company publishes at least as many fiction books as nonfiction books. This means: \( f \geq n \).

Step 5: System of Inequalities

Combining all the expressions, the system of inequalities is: \[ \begin{align*} f + n & \leq 100, \ n & \geq 20, \ f & \geq n \end{align*} \]

Step 6: Graph the Inequality System

To graph this system, draw the following:- A line for \( f + n = 100 \) and shade below the line, since we're interested in the region where \( f + n \leq 100 \).- A horizontal line at \( n = 20 \) and shade above the line for \( n \geq 20 \).- The line \( f = n \) and shade above the line (right of \( n \) in terms of intersection with axes) for \( f \geq n \).The feasible region is bounded where all shading overlaps.

Key Concepts

Linear Inequalities

Linear inequalities are expressions that show the relationship between two expressions where one side is not strictly equal but rather less than or greater than the other. In the context of publishing books, these inequalities help define the different constraints and requirements, such as the total number of books being published each year.

In the exercise, we have several linear inequalities:

• \( f + n \leq 100 \): This inequality represents the constraint that the total number of fiction (\( f \)) and nonfiction (\( n \)) books published cannot exceed 100. This is a classic example of an inequality that sets a boundary for a sum of terms.
• \( n \geq 20 \): This inequality ensures that at least 20 nonfiction books must be published, setting a lower limit.
• \( f \geq n \): This constraint indicates that the number of fiction books must be at least as many as nonfiction, adding an additional layer to our system of equations.

Linear inequalities like these create a boundary or a range of solutions, where any combination of \( f \) and \( n \) within these limits will satisfy the conditions for publishing books. They help in finding feasible solutions through calculations and graphs.

Graphing Inequalities

Graphing inequalities is a powerful visual tool used to find solutions that satisfy a set of conditions. When working with two variables, such as in our book publishing problem, the graph can illustrate the feasible region where all inequalities are satisfied.

To graph the inequalities in our exercise, we:

• First, draw the line \( f + n = 100 \) on a graph. This line represents the equation where the sum of fiction and nonfiction books is exactly 100. To represent all solutions where the number of books is less than or equal to 100, we shade the area below this line.
• Next, draw a horizontal line at \( n = 20 \). This line signifies the minimum number of nonfiction books. Shade above this line to represent the solutions where \( n \geq 20 \).
• Finally, graph the line \( f = n \). Here, you shade the region above this line (to the right of the intersection point) since we want \( f \geq n \), meaning more or equal fiction to nonfiction books.

The overlapping shaded region on the graph, created from these combined conditions, shows all possible combinations of fiction and nonfiction books that satisfy all the publishing constraints.

Variables in Mathematics

In mathematics, variables play a crucial role in formulating expressions, equations, and inequalities. A variable acts as a placeholder for unknown values that we solve for within various constraints and conditions.

In our book publishing scenario, we use two variables:

• \( f \): Representing the number of fiction books published. This variable captures all possible quantities of fictional works that fall within the constraints of the publishing company.
• \( n \): Denoting the number of nonfiction books. This variable provides a way to quantify and manage the company's nonfiction output while considering minimum requirements.

These variables help us convert real-world situations into mathematical models. By defining clear variables, we can develop precise inequalities, manipulate these in calculations, and finally use them in graphs to find viable solutions. Through variables, complex conditions involving any number of factors can be systematically approached and solved, showcasing their fundamental role in mathematics.