Problem 25 - 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. Write the model equation for the cost of renting a truck with plan B.

Step-by-Step Solution

Answer

The model equation for Plan B is \( C = 100 + 0.05m \).

Step 1: Identify Components of Plan B

For Plan B, the truck rental agency charges an initial fee of $100 per week. Additionally, there is a cost associated with miles driven, which is $0.05 per mile.

Step 2: Define Variables

We need to choose a variable to represent the number of miles driven. Let's define the variable \( m \) as the number of miles driven in a week.

Step 3: Set Up the Cost Equation for Plan B

Use the fixed weekly charge and the variable mileage charge to write the equation. The cost \( C \) is equal to the weekly charge plus the mileage charge: \( C = 100 + 0.05m \).

Step 4: Interpret the Model Equation

The equation \( C = 100 + 0.05m \) represents the total cost of renting a truck for a week under Plan B, where 100 is the fixed weekly cost and \( 0.05m \) represents the cost for the miles driven.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebraic modeling. They describe a straight line when graphed and are used to express relationships between two variables. In the context of truck rental plans, a linear equation models the total cost based on the number of miles driven. The general form of a linear equation is \( y = mx + b \), where \( y \) is the dependent variable (total cost), \( m \) is the slope (rate per mile in this case), \( x \) is the independent variable (number of miles), and \( b \) is the y-intercept (fixed weekly charge).
When you write the equation for a real-world scenario, like Plan B, you have:

• The fixed cost: \( b = 100 \) dollars per week
• The variable cost per mile: \( m = 0.05 \)

Thus, the linear equation is \( C = 100 + 0.05m \). Here, the cost \( C \) changes with \( m \) while the rest remain constant. This linear equation helps predict costs for varying numbers of miles driven.

Cost Analysis

Cost analysis is about understanding and comparing costs, often to determine which option is more economical. For truck rentals, it involves evaluating different plans to decide which costs less based on anticipated mileage.
For Plan B, to do a cost analysis, consider both the fixed weekly cost and the cost per mile. The fixed cost of renting the truck under Plan B is \( \\(100 \), and you add \( \\)0.05 \) per mile. When comparing plan costs, you add this variable component to calculate total expenses:

• Fixed Cost: \( \\(100 \)
• Cost per mile: \( \\)0.05 \times \text{number of miles} \)

Evaluate this for different mileages and compare with other plans. This helps in decision-making to choose a cost-effective plan based on mileage estimates. Cost analysis uses linear equations to predict how costs escalate with more miles driven.

Variables in Algebra

Variables are symbols, usually letters, used to represent unknown values or values that can change. In algebra, they're crucial for creating models of real-world situations. In the truck rental scenario, we use the variable \( m \) to denote the number of miles driven.
Understanding variables helps in setting up equations that accurately represent the situation. Using variables allows us to calculate different outcomes by simply changing the variable's value. For example:

• If \( m = 200 \), substitute in \( C = 100 + 0.05m \), to find \( C \)
• This becomes \( C = 100 + 0.05 \times 200 \), which calculates to \( C = 110 \)

By defining \( m \), we automate calculations for any mileage. This principle of using variables is foundational in algebra, allowing flexible modeling and problem-solving in many fields.