Problem 132 - Algebra and Trigonometry

Question

Exercises \(131-133\) will help you prepare for the material covered in the next section. A telephone texting plan has a monthly fee of \(\$ 20\) with a charge of \(\$ 0.05\) per text. Write an algebraic expression that models the plan's monthly cost for \(x\) text messages.

Step-by-Step Solution

Answer

The algebraic expression to model the monthly cost for \(x\) text messages is \$20 + \$0.05*x.

Step 1: Identify the Fixed Cost

The fixed cost is the cost that is the same every month, regardless of the number of text messages. Here, the fixed cost is \$20 per month.

Step 2: Identify the Variable Cost

The variable cost is the cost that changes based on the number of text messages. In this case, it is \$0.05 per text message.

Step 3: Write the Algebraic Expression

To calculate the total monthly cost, you combine the fixed and variable costs. The variable cost is multiplied by the number of text messages, represented by \(x\), and the fixed cost is added. So, the expression is \$20 + \$0.05*x.

Key Concepts

Understanding Fixed and Variable Costs

When looking at financial expenditures, costs are often categorized into two main types: fixed costs and variable costs. Fixed costs are expenses that do not change regardless of the activity level. For example, rent for a business space or a subscription fee. These costs remain consistent in total no matter how much is produced or consumed.

On the other hand, variable costs fluctuate with the level of output or activity. These costs increase as activity increases and decrease as activity decreases. Examples include the cost of raw materials for manufacturing or, as in our exercise, the charge per text message in a mobile phone plan.

In the given exercise, the fixed cost is represented by the monthly fee of \(20 for the texting plan. It's a set amount that the customer must pay each month, regardless of how many text messages are sent. Conversely, the variable cost is presented as \)0.05 per text, which will vary based on the number of texts sent during the month. Understanding these two types of costs is crucial in both business budgeting and algebraic modeling of real-world scenarios.

Introduction to Algebraic Modeling

Algebraic modeling involves using algebraic expressions to represent real-world situations mathematically. This practice converts the given information into mathematical symbols and formulas, which can then be manipulated to find solutions to various problems. In the context of our exercise, we use algebraic modeling to represent the cost structure of a telephone texting plan.

The process starts by identifying all the components involved: the fixed cost of the plan, the variable cost per each unit of activity (in this case, per text message), and the quantity of the activity, which is the unknown variable we represent as x. By constructing an algebraic expression, we create a model that can calculate the total monthly cost based on any number of text messages.

Components of Algebraic Modeling

• Define the constants (fixed costs).
• Identify the variable factors (variable costs).
• Variable representation (using symbols like x).
• Construction of an algebraic expression (formula).

Algebraic modeling enables us to calculate outcomes under various scenarios without manually computing each possible situation, making it a powerful tool in both learning and real-life applications.

Working with Linear Expressions

A linear expression in algebra is a mathematical statement that describes a relationship involving a constant term and one or more terms each being multiplied by a constant, which are raised to the first power. In simpler terms, it's an algebraic expression where the variable does not have an exponent other than one, which results in a straight-line graph when plotted.

In our exercise, the linear expression we create to represent the total monthly cost of a texting plan is based on the number of text messages sent. It has two components: a fixed cost (\(20), which is our constant term, and a variable cost (\)0.05) that is multiplied by the number of messages (represented as x). Written out, the linear expression is:
\[\begin{equation} Total\underline{\phantom{xxx}}Cost = Fixed\underline{\phantom{xxx}}Cost + (Variable\underline{\phantom{xxx}}Cost \times Number\underline{\phantom{xxx}}of\underline{\phantom{xxx}}Texts) \ Total\underline{\phantom{xxx}}Cost = \(20 + (\)0.05 \times x) \
\text{where } x \text{ is the number of text messages.} \end{equation}\]

Characteristics of Linear Expressions

• No exponents on the variable (except the implicit 1).
• The graph of a linear expression is always a straight line.
• Can consist of constant terms (like the fixed cost) and coefficients of the variable terms (like the charge per text).

Linear expressions are fundamental in algebra and are frequently used to model a direct proportional relationship between two variables, such as cost and quantity.