Problem 39 - Intermediate Algebra

Question

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The cost each semester is the sum of \(\$ 13\) times the number of units taken and a student services fee of \(\$ 24\)

Step-by-Step Solution

Answer

The model is \( C = 13u + 24 \).

Step 1: Define the Variables

First, define the variables for the problem. Let \( u \) represent the number of units taken, and let \( C \) represent the total cost for the semester.

Step 2: Identify the Constants

Next, identify the constants from the problem statement. The cost per unit is \( \\( 13 \) and the student services fee is \( \\) 24 \). These values are fixed, regardless of the number of units taken.

Step 3: Write the Mathematical Model

Using the defined variables and constants, write the mathematical model that represents the total cost. The total cost \( C \) is given by the formula: \[ C = 13u + 24 \] This equation shows that the cost is \( \\( 13 \) times the number of units, plus an additional \( \\) 24 \) for the student services fee.

Key Concepts

Cost Analysis

In the realm of mathematical modeling, **cost analysis** is crucial for understanding and predicting financial outcomes. In the context of the exercise, cost analysis involves breaking down the components of the total semester cost for a student. This includes summing fixed and variable costs.

* **Fixed Costs**: These do not change, regardless of other factors. Here, the student services fee of \( 24 \) is constant, no matter how many units a student takes.
* **Variable Costs**: These fluctuate according to certain variables. In this example, the cost per unit is \( 13 \), which means it increases with each unit taken.

By analyzing these components, students can see how each part contributes to the total cost. Understanding the role of fixed and variable costs is essential in budgeting and financial decision-making.

Variable Definition

Defining **variables** is a fundamental part of mathematical modeling and solving such problems effectively. Variables are symbols, often letters, used to represent unknown values or quantities that can vary.
Here, we defined:

• \( u \) as the number of units taken by the student, representing the variable component. It can change, influencing the resulting total cost.
• \( C \) as the total semester cost, which is the outcome variable we want to compute using our model.

Defining variables is step one in establishing a base for constructing mathematical models. Clear variable definitions allow you to clearly communicate and manipulate equations.

Mathematical Equations

A mathematical equation is a tool for expressing relationships between variables and constants. They use a combination of numbers, symbols, and operators. The model we crafted for this exercise is a **linear equation**: \[ C = 13u + 24 \] This equation translates the relationship between the units taken and the total costs into a compact mathematical form. It tells us that the total cost \( C \) is derived by multiplying the number of units \( u \) by the unit cost \( 13 \), then adding the fixed service fee of \( 24 \).

Mathematical equations are powerful for modeling real-life situations as they offer a precise, straightforward mechanism to calculate outcomes based on variable inputs.