Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $$\$ 75 / w$$ k plus $$\$.10 / \mathrm{mi}$$ driven. Plan \(\mathrm{B}\) 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 is \( C = 100 + 0.05x \).

Step 1: Identify the Fixed Weekly Cost for Plan B

Plan B has a fixed weekly cost of $100. This means that no matter how many miles are driven, $100 is the starting amount for the cost of renting.

Step 2: Identify the Variable Cost Per Mile for Plan B

The variable cost in Plan B is determined by the distance driven. Plan B charges $0.05 for each mile driven.

Step 3: Define the Variables in the Equation

To construct an equation, we introduce the variable \( x \) to represent the number of miles driven in a week.

Step 4: Write the Model Equation for Plan B

Combine the fixed cost and the variable cost to form the equation. The total cost, which we will denote as \( C \), is the fixed weekly cost plus the cost per mile times the number of miles driven:\[ C = 100 + 0.05x \]

Step 5: Verify the Model Equation

Check the equation to ensure it represents all factors: the \(100 fixed cost and the \)0.05 per mile. The equation \( C = 100 + 0.05x \) correctly represents these components.

Key Concepts

fixed and variable costs

When renting a truck, the cost you pay is composed of two main parts: fixed costs and variable costs. Understanding these can help you figure out your total expenses easily.
Fixed costs remain the same, no matter how much or how little you use the service. For Plan B, this includes a flat fee of $100 each week you rent the truck. This fixed cost guarantees you have the truck available, regardless of how much you drive it.
Variable costs, on the other hand, depend on your usage. In this case, Plan B charges $0.05 for every mile you drive. If you drive more, your cost will increase, and if you drive less, it will decrease.

• Fixed cost for Plan B: $100 per week, unchanging.
• Variable cost for Plan B: $0.05 per mile driven.

Knowing the difference between these two types of costs is crucial for managing expenses effectively.

linear equations

A linear equation is a way to describe a relationship between two variables using a straight line. This type of equation is commonly used to model real-world scenarios, such as calculating the cost of services.
For Plan B, the relationship between the total cost and the number of miles driven is linear. The equation for Plan B is: \[ C = 100 + 0.05x \]This equation helps us predict the cost by using two components:

• The fixed component (\(100 weekly fee)
• The variable component (\)0.05 per mile driven)

Each part is added together to give you the total cost, depicted by a straight line when graphed. Understanding how to set up and solve linear equations is key in figuring out costs, profits, and other business calculations.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication) that represent a specific value or function.
In the context of analyzing truck rental costs, the expression for Plan B is \[ 100 + 0.05x \]. This can be broken down as follows:

• The number 100 represents the fixed weekly cost, a constant term in the expression.
• The term \(0.05x\) represents the variable cost per mile, where \(x\) is the number of miles driven.

By understanding how to read and construct algebraic expressions, you can model various scenarios and predict outcomes, like how much a rental will cost depending on mileage. Mastery of algebraic expressions can greatly aid in solving practical problems and making informed financial decisions.