Problem 109 - Introductory Algebra for College Students

Question

A car can be rented from Continental Rental for 80 dollars per week plus 25 cents for each mile driven. How many miles can you travel if you can spend at most 400 dollars for the week?

Step-by-Step Solution

Answer

The maximum number of miles you can travel for $400 is 1,280 miles.

Step 1: Understand the cost structure

The cost of renting a car from Continental Rental constitutes a flat fee of $80 per week and a variable cost of $0.25 per mile driven.

Step 2: Setting up the equation

If the budget is a maximum of $400 and the weekly flat fee is a known value, represent the cost of the car rental as: \( 80 + 0.25*m = 400 \) where m is the number of miles that can be driven.

Step 3: Solving for 'm'

Isolate 'm' by first subtracting 80 from both sides to get: \( 0.25*m = 400 - 80 \). Then divide both sides by 0.25 to solve for 'm': \( m = \frac{(400 - 80)}{0.25} \). Calculate the right-hand side to find the value of 'm'.

Key Concepts

Cost Analysis

When we talk about cost analysis in the context of this problem, we're examining the total expenses involved in renting a car. This includes both fixed and variable costs.

• Fixed Costs: For Continental Rental, the fixed cost is $80 per week. No matter how many miles you drive, this amount remains unchanged.
• Variable Costs: These costs change depending on how much you use the service. Here, it's $0.25 per mile driven. The more miles you drive, the more you pay.

Understanding these two types of costs is crucial for planning your budget and keeping expenses under control. You'll know exactly how much you will spend, and forecast your needs accordingly. For example, if you plan a long trip, the variable costs will become more significant, influencing your total expense drastically.

Problem Solving

Problem-solving involves systematic approaches to understanding and solving a given problem. Here’s how you can apply it to our car rental situation:

• **Identify the elements**: You must first understand the components affecting the total cost. Here it's the flat fee and the per-mile charge.
• **Set a goal**: Decide on your budget limit ($400 in this scenario).
• **Formulate a plan**: Establish an equation or strategy to see how many miles you can travel without exceeding your budget.

In our example, the problem-solving process involves setting up an equation to represent the financial constraint. This will direct you to calculate the maximum number of miles you can drive to not surpass your budget. Keeping a structured strategy ensures you can handle the problem efficiently and correctly.

Linear Equations

Linear equations are mathematical expressions that represent relationships with constant rates of change. They follow the form \( ax + b = c \), where \(a\), \(b\), and \(c\) are constants. Let's relate this to our car rental scenario:

• **Equation Setup**: We set up our equation as \(80 + 0.25*m = 400\). Here, 80 and 0.25 are constants, representing the fixed and variable costs respectively.
• **Solving the Equation**: To solve for \(m\), the number of miles, rearrange the equation. Subtract the fixed cost from your total budget, and then divide by the per mile cost.

Using algebraic principles, these equations allow you to solve for unknown quantities, making them essential in predicting and planning within constrained scenarios such as budgeting.