Problem 53 - College Algebra with Corequisite Support

Question

You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $$\$ 19.95$$ plus 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable?

Step-by-Step Solution

Answer

You need to use more than 133 minutes for the second plan to be preferable.

Step 1: Define the Variables

Let's define \( x \) as the number of minutes used in a month. Our goal is to find the value of \( x \) for which the second plan becomes more cost-effective than the first plan.

Step 2: Write the Cost Equation for Each Plan

For the first plan, the cost equation is given by \( C_1 = 0.26x \), where \( x \) is the number of minutes. For the second plan, the cost equation is \( C_2 = 19.95 + 0.11x \), which includes a monthly fee of \$19.95 and charges 11 cents per minute.

Step 3: Set Up an Inequality

To determine when the second plan is cheaper, set the cost of the second plan less than the first plan: \( 19.95 + 0.11x < 0.26x \).

Step 4: Solve the Inequality

Subtract \( 0.11x \) from both sides to get \( 19.95 < 0.15x \). Then, divide both sides by 0.15 to isolate \( x \): \( x > \frac{19.95}{0.15} \).

Step 5: Calculate the Solution

Perform the division: \( x > \frac{19.95}{0.15} \approx 133 \). Therefore, you need to use more than 133 minutes.

Key Concepts

Algebraic Equations

Algebraic equations are foundational in solving many real-world problems, including those involving costs. An algebraic equation is a mathematical statement that shows the equal relationship between two expressions. It can involve variables, constants, and arithmetic operations. In our exercise, we used algebraic equations to express the cost of two different cell phone plans. By defining the cost of each plan with a formula, using symbols such as \( x \) to represent the number of minutes, we can creatively compare and contrast the expenses.
The formula for each plan was developed based on the given rate of cost per minute.

• First Plan: The equation \( C_1 = 0.26x \) shows that the cost is directly proportional to the number of minutes, \( x \), with a rate of 26 cents per minute.
• Second Plan: The equation \( C_2 = 19.95 + 0.11x \) represents a fixed monthly charge plus an additional variable charge based on the minutes used.

Through algebraic equations, we can effectively communicate these cost relationships and prepare them for further analysis such as comparisons and optimizations.

Cost Analysis

Cost analysis helps us to identify the most economical options among several alternatives by comparing costs under similar conditions. In this exercise, we used cost analysis to determine when the second phone plan would be more cost-effective. We created an inequality to reflect the relationship between the two cost conditions.
The goal was to find the point at which the second plan became cheaper:

• The inequality \( 19.95 + 0.11x < 0.26x \) depicts when the cost of the second plan is less than the first plan.
• This analysis included breaking down the cost elements, such as fixed fees and per-minute rates, and assessing them over the expected usage.

By using cost analysis through inequalities, we can make informed decisions regarding our choice, ensuring that our selections are guided by financial efficiency and understanding of the relevant factors.

Linear Functions

Linear functions play a crucial role in modeling relationships and variations between quantities. A linear function is based on the principle of a constant rate of change, which is represented graphically by a straight line in a coordinate plane. In our case, both prepaid plans represent linear cost functions.
Each plan's cost function can be described by the general form \( y = mx + b \), where

• \( m \) is the slope, representing the per-minute cost.
• \( b \) is the y-intercept, representing fixed costs or fees.

While the first plan's linear equation \( C_1 = 0.26x \) has no fixed fee, indicating that the line passes through the origin, the second plan \( C_2 = 19.95 + 0.11x \) starts at \( 19.95 \), climbing up as usage increases. Linear functions are immensely useful for quick and easy identification of trends and costs, providing actionable insights in decision-making processes.