Problem 132 - College Algebra

Question

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 that represents the monthly cost is \(C = 20 + 0.05x\)

Step 1: Identify the constant

The constant cost is the monthly fee that needs to be paid regardless of the number of text messages. In this scenario, it is given as $20.

Step 2: Identify the variable cost

The variable cost depends on the quantity of a certain variable - in this case the number of text messages. Each message costs an additional $0.05.

Step 3: Structure the algebraic expression

To find the total cost \(C\), combine the constant cost with the product of the variable and the cost per unit. This results in the formula \(C = 20 + 0.05x\), where \(x\) represents the number of messages.

Key Concepts

Variable Costs

Variable costs are the type of expenses that change based on the amount of usage or production. In our example of a telephone texting plan, think of variable costs as the extra expenses that depend on the number of text messages you send each month. For each text message, there is a cost of \( \(0.05 \). This means if you send one message, you pay \( \)0.05 \). If you send 100 messages, the variable cost becomes \( 0.05 \times 100 = $5 \). The more messages you send, the higher your variable cost will be, as it directly depends on the number of texts.

• Variable costs increase or decrease with activity level.
• In this scenario, it's given by \( 0.05x \), where \( x \) is the number of text messages.
• Understanding variable costs helps in predicting how expenses will change with usage.

By adding these variable costs to the constant charges, you can model the total expense for the month.

Constant Terms

Constant terms are fixed costs or fees that do not change regardless of how much you use the service or product. In the context of the texting plan, the constant term is the monthly fee of \( \(20 \), which you pay irrespective of whether you send zero or many messages.Constant terms allow for predictable budgeting, as you know these costs will remain the same no matter the circumstances.

• Always the same amount; unaffected by quantity.
• Example: In this problem, the \( \)20 \) monthly fee.
• They confirm that expenses will cover some level of service.

So, even if you don't send any text messages, you'll still have this fixed cost every month. This is essential to factor in when calculating total expenses.

Modeling Costs

Modeling costs involves creating an equation that represents how different costs add up over time. In our exercise, the goal is to find out the total monthly cost of the texting plan. To do this, you combine both the variable costs and the constant terms into a single algebraic expression.The expression \( C = 20 + 0.05x \) allows you to calculate the monthly cost \( C \) by adding the fixed monthly fee of \( $20 \) to the variable cost of text messages, \( 0.05x \). Here, \( x \) represents the number of texts.