Problem 26 - College Algebra

Question

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges \(\$ 75 / \mathrm{wk}\) . plus \(\$ .10 / \mathrm{mi}\) driven. Plan B charges \(\$ 100 / \mathrm{wk}\) plus \(\$ .05 \mathrm{mi}\) driven. Find the number of miles that would generate the same cost for both plans.

Step-by-Step Solution

Answer

The plans cost the same at 500 miles driven.

Step 1: Understand the Problem

We are given two pricing plans for renting a truck. Plan A costs $75 per week plus $0.10 per mile driven. Plan B costs $100 per week plus $0.05 per mile driven. We need to find the number of miles driven that makes both plans equal in cost.

Step 2: Set Up Equations for Costs

Let \( m \) be the number of miles driven. For Plan A, the total cost \( C_A \) is given by\[ C_A = 75 + 0.10m \]For Plan B, the total cost \( C_B \) is given by\[ C_B = 100 + 0.05m \]

Step 3: Set the Equations Equal

To find the number of miles where the costs are equal, set the two cost equations equal to each other:\[ 75 + 0.10m = 100 + 0.05m \]

Step 4: Solve for the Number of Miles

First, subtract \(0.05m\) from both sides of the equation:\[ 0.10m - 0.05m = 100 - 75 \]This simplifies to:\[ 0.05m = 25 \]Now, solve for \(m\) by dividing both sides by \(0.05\):\[ m = \frac{25}{0.05} = 500 \]

Step 5: Conclusion

The number of miles that will make both plans equal in cost is 500 miles.

Key Concepts

Understanding Cost Comparison

When deciding between two or more options, it's essential to compare the associated costs effectively. Cost comparison involves analyzing different plans, prices, or strategies to determine the best choice based on financial factors. In the context of the given exercise, cost comparison is used to analyze two truck rental plans: Plan A and Plan B.

To compare them accurately, you need to understand the fixed and variable components of each plan.

• **Fixed Costs**: These are costs that remain constant regardless of usage, like the flat rate per week ( $75 for Plan A and $100 for Plan B).
• **Variable Costs**: These are costs that change based on usage, such as the cost per mile driven ($0.10 for Plan A and $0.05 for Plan B).

By identifying and comparing these elements, you can calculate the total cost for each plan over the same usage level, which allows for an informed decision about which plan is more cost-effective at different driving distances. The goal here is to find the breakeven point where both plans cost the same, which is determined by the number of miles driven.

Linear Equations in Cost Analysis

Linear equations can simplify and solve real-world problems by modeling relationships in a straightforward mathematical form. In this exercise, linear equations represent the total cost of each rental plan based on miles driven.

• **Plan A Equation**: This is represented by the equation \(C_A = 75 + 0.10m\), reflecting a base cost plus a cost per mile.
• **Plan B Equation**: Similarly, for Plan B, it's \(C_B = 100 + 0.05m\), following the same pattern.

Using these equations, the process involves expressing costs as linear functions of the variable \(m\), which stands for miles driven. When you set these equations equal to each other, it indicates the point at which the costs of both plans are identical.

The beauty of linear equations is their simplicity and versatility. They help in visualizing how small changes in miles driven impact overall costs. By solving the equation \(75 + 0.10m = 100 + 0.05m\), you determine at what mileage both plans yield the same expenditure.

Steps for Solving Problems

There are specific steps involved in methodically solving a problem, which makes the process easier and more systematic. Here, the focus is on solving for the breakeven mileage where two plans cost the same.

• **Understand the Problem**: Recognize what is being asked. Here, it is about finding equal costs between two plans.
• **Set Up Equations**: Translate the word problem into mathematical equations. For this problem, we use linear equations to represent each plan's cost.
• **Set Equations Equal**: Equate the two costs to find where they are the same.
• **Solve the Equation**: Simplify and solve for the unknown. For this scenario, solving \(0.10m - 0.05m = 25\) determines \(m\), the equal mileage.
• **Conclusion**: Interpret the results correctly. The solution means 500 miles is the point where plans are equally cost-effective.

Approaching problems with a structured method gives clarity and helps resolve even complex-looking problems. Solving for equal costs using these steps not only helps in this particular scenario but also builds skills applicable to many areas of math and daily life decision-making.