Problem 64 - College Algebra

Question

A basic cellular package costs \(\$ 20 / \mathrm{mo}\) . for 60 \(\mathrm{min}\) of calling, with an additional charge of \(\$ 0.30 / \mathrm{min}\) beyond that time. The cost formula would be \(C=\$ 20+.30(x-60) .\) 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 up to 159 minutes.

Step 1: Understand the Equation

The equation for the cost of the cellular package is given as \( C = 20 + 0.30(x - 60) \). Here, \( C \) represents the total cost in dollars, and \( x \) is the total number of minutes used in a month. The first part, \( 20 \), is the base cost for the first 60 minutes of calling. The second part, \( 0.30(x - 60) \), is the additional cost for the minutes beyond the initial 60 minutes.

Step 2: Set up the Inequality

Since you want to keep the bill lower than \( \\(50 \), we set up the inequality: \[ 20 + 0.30(x - 60) < 50 \] This inequality states that the total cost should be less than \( \\)50 \).

Step 3: Simplify the Inequality

Solve the inequality step-by-step: 1. Subtract 20 from both sides: \[ 0.30(x - 60) < 30 \] 2. Divide both sides by 0.30 to isolate \( (x - 60) \): \[ x - 60 < 100 \]

Step 4: Solve for x

Finally, solve for \( x \) by adding 60 to both sides: \[ x < 160 \] So, \( x \) can be at most 159, as we want to keep the cost strictly under \( \$50 \).

Step 5: Conclusion

Summarize the result: To ensure the bill remains under \( \$50 \), you can use up to 159 minutes of calling in total. This includes the base 60 minutes and an additional 99 minutes at the extra rate.

Key Concepts

Understanding Inequalities

In mathematics, an inequality compares two values or expressions, showing that one is either less than, greater than, or (in some cases) equal to the other. It provides a way to express limits or boundaries.
In the context of the given problem, the inequality is used to express that we need the total cellular cost to be less than $50.
Taking the cost formula, which includes both fixed and variable parts, we frame the inequality like this:

• The base cost: $20 for the first 60 minutes.
• The extra cost: $0.30 per additional minute.
• Total cost: Must be under $50.

This sets up a mathematical expression which helps us solve for the maximum calling minutes possible without exceeding the budget.

Choosing the Right Cellular Package

Cellular packages often blend a base cost with additional charges for extra usage. Understanding these packages involves calculating potential expenses based on expected usage.
In our scenario, the basic package offers 60 calling minutes for $20 monthly. But if you talk beyond those 60 minutes, extra charges apply. This structure is common across many phone plans.

• The base fee provides a predictable monthly charge.
• Extra charges can increase the cost based on usage.
• Setting a budget limit (like under $50) helps manage expenses.

Learning how to balance these factors is crucial for choosing the right plan to fit your needs.

Calculating Calling Minutes

Understanding calling minutes in a cellular plan involves looking at both base and extra usage. The task is to determine how much you can talk without exceeding your budget limit.
In this particular problem, we start with a base of 60 minutes. After that, each additional minute comes at a cost, which is tagged as extra charges. Solving involves:

• Knowing your base limit, such as 60 minutes, included in the package.
• Computing how many more minutes you can use while staying within the total allowable cost.
• Setting up algebraic expressions or inequalities to find this maximum number of minutes.

This ensures you stay within budget and make the most of your cellular plan.

Finding the Algebraic Solution

An algebraic solution requires breaking down the given inequality to find the desired variable, in this case, the number of calling minutes.
Based on the cost equation given as: \( C = 20 + 0.30(x - 60) \), the steps to solve for maximum minutes are clear:

• Start by setting up the inequality with the budget constraint.
• Simplify the inequality step by step by isolating terms.
• Calculate and interpret the solution effectively.

Ultimately, by adding and subtracting from both sides and manipulating the inequality, we determined that you can talk up to 159 minutes. This includes the initial 60 minutes and up to 99 extra minutes.