Problem 41 - Prealgebra

Question

Use the following information. (Lesson 1-3) Cornet Cable charges 32.50 dollars a month for basic cable television. Each premium channel selected costs an additional 4.95 dollars per month. Write an expression to find the cost of a month of cable service.

Step-by-Step Solution

Answer

The total cost expression is \( C = 32.50 + 4.95p \).

Step 1: Understanding the Problem

First, let's identify the two main charges in the problem. We have a basic cable service charge and a cost for each premium channel. The basic service is a flat rate of $32.50 per month, and each additional premium channel costs $4.95 per month.

Step 2: Define the Variables

To write an expression for the total cost, we need a variable to represent the number of premium channels selected. Let's use \( p \) to denote the number of premium channels.

Step 3: Write the Expression

The total cost of cable service will be the sum of the basic cable service and the cost of the premium channels. For the basic service, the cost is \(32.50. For the premium channels, the cost is \( 4.95p \) since each channel costs \)4.95. Thus, the expression for the total cost \( C \) can be written as: \[ C = 32.50 + 4.95p \]

Key Concepts

Variables in Algebraic Expressions

Variables play a huge role in mathematics, especially in algebraic expressions. They are symbols, usually letters, that represent unknown or changeable values. In our exercise, we need to calculate the total cost of cable service. To do this, we use a variable.

• This variable will represent the number of premium channels a customer might choose.
• For simplicity, we've chosen the letter \( p \) as our variable. You can pick any letter, but \( p \) makes sense here because it reminds us of the word 'premium'.

With this variable in place, it's easy to make our calculations flexible. You can substitute any number into \( p \) to find out how much the monthly charge would be for any possible number of premium channels.

Understanding Linear Equations

Linear equations are among the building blocks of algebra. These are equations where the variables are raised only to the power of one, resulting in a straight line graph when plotted. In our case, the expression \[ C = 32.50 + 4.95p \] is a linear equation.Here’s why:

• The equation includes a fixed cost—this is the base price of \( 32.50 \), which doesn't change regardless of the number of premium channels.
• To this fixed amount, we add \( 4.95p \), representing the cost per additional premium channel, multiplied by the number of such channels.

This structure makes the equation linear, as you essentially create a straight line of possible costs versus the number of premium channels, based on changes to \( p \).Linear equations like this one are great for modeling real-world situations where costs increase proportionally with usage.

The Importance of Cost Calculation

Cost calculation is crucial, whether it's for budgeting personal expenses or managing company finances. In this exercise, we calculate the total cost of cable service, which includes both a fixed basic charge and variable charges for premium channels. Here's how each component plays a role:

• Fixed Costs: The basic cable package costs \( 32.50 \) regardless of extra channels. This provides a predictable base for monthly budgeting.
• Variable Costs: Each premium channel adds \( 4.95 \) to the bill. The total additional cost depends on how many channels are selected.

Understanding these costs helps one decide how many premium channels to afford without overspending. By applying the algebraic expression \( C = 32.50 + 4.95p \), anyone can quickly compute their expected bill. This is handy when managing overall expenses and staying on top of a budget.