Problem 64 - College Algebra with Corequisite Support

Question

A basic cellular package costs $$\$ 20 / \mathrm{mo} .$$ for \(60 \mathrm{~min}\) of calling, with an additional charge of $$\$ .30 /$$ min beyond that time.. The cost formula would be $$C=\$ 20+.30(x-60) . \text { If you }$$ have to keep your bill lower than $$\$ 50,$$ what is the maximum calling minutes you can use?

Step-by-Step Solution

Answer

You can use a maximum of 159 minutes.

Step 1: Understanding the Problem

The cost formula for the basic cellular package is given by \( C = 20 + 0.30(x - 60) \), where \( x \) is the total minutes of call. We need to find the maximum \( x \) such that the total cost is less than $50.

Step 2: Setting Up the Equation

To keep the bill under $50, we need \( C < 50 \). Substitute the cost formula into this inequality: \( 20 + 0.30(x - 60) < 50 \).

Step 3: Isolating the Variable

Subtract 20 from both sides of the inequality: \( 0.30(x - 60) < 30 \).

Step 4: Solving for x

Divide both sides by 0.30 to isolate \( x - 60 \): \( x - 60 < 100 \). Next, add 60 to both sides to solve for \( x \): \( x < 160 \).

Step 5: Conclusion

The maximum calling minutes you can use to keep the bill under $50 is 159 minutes, since \( x \) must be an integer and less than 160.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators that represent a value. In this exercise, the algebraic expression used is the cost formula:

• Cost Formula: The expression given is \( C = 20 + 0.30(x - 60) \).
• Components: Here, 20 is the fixed monthly cost, 0.30 is the cost per minute over 60 minutes, and \( x \) is the total calling minutes.

Algebraic expressions need to accurately represent real-world scenarios to solve problems effectively. This involves understanding terms like constants (e.g., 20), coefficients (e.g., 0.30), and variables (e.g., \( x \)).
The essence of this exercise is to transform a real-world situation into a mathematical expression and then manipulate this expression to derive useful information, such as the maximum calling minutes you can use before exceeding a certain cost.

Linear Inequalities

Linear inequalities are algebraic expressions that involve inequality signs such as <, >, ≤, or ≥. They represent conditions or restrictions in a problem statement that must be met. In this case, the inequality is used to ensure the total cost remains below a certain amount:

• Inequality Set Up: The given condition is \(20 + 0.30(x - 60) < 50\), which represents that the total cost \( C \) must be less than \(50.
• Solving Inequalities: To solve, manipulate the inequality in the same way you would for an equation. Key steps include isolating the variable and keeping the inequality balanced through operations such as addition, subtraction, multiplication, or division.

While solving, it's crucial to ensure that any operations performed on inequalities preserve the inequality's direction, especially reversing the sign when multiplying or dividing by negative numbers. Here, however, we don't multiply or divide by negatives, making the solution straightforward. The resulting inequality helps to determine the maximum permissible total minutes of calling before surpassing the \)50 cost limit.

Problem Solving

Problem solving involves a structured approach to finding solutions to specific challenges. In exercises involving inequalities, such as this one, understanding the problem thoroughly is crucial:

• Understand the Problem: Grasp the relationship between all components involved, such as the fixed cost, variable minutes, and additional charges.
• Formulate the Inequality: Translate this understanding into a mathematical model, like the cost inequality \(20 + 0.30(x - 60) < 50\).
• Solve the Inequality: Systematically solve the inequality through appropriate algebraic manipulations as shown in the step-by-step solution.
• Interpret the Solution: Once an inequality solution is found, interpret it in the context of the original problem, ensuring the practical application of the mathematical result.

Successful problem solving requires critical thinking and step-by-step reasoning to ensure each part of the original exercise is addressed. Ultimately, the solution \( x < 160 \) reveals that at most 159 calling minutes keeps the cost under $50, effectively solving the problem.