Problem 32 - Intermediate Algebra

Question

Use linear functions. The Rent-Me Car Rental charges \(\$ 35\) per day plus \(\$ 0.32\) per mile to rent a car. Determine a linear function that can be used to calculate daily car rentals. Then use that function to determine the cost of renting a car for a day and driving 150 miles; 230 miles; 360 miles; 430 miles.

Step-by-Step Solution

Answer

The costs are $83, $108.60, $150.20, and $172.60 for 150, 230, 360, and 430 miles respectively.

Step 1: Define the Variables

Let's define the variables to help us formulate a function. Let \( C \) be the total cost of renting the car, \( d \) be the fixed daily charge, and \( m \) be the number of miles driven.

Step 2: Write the Cost Function

The total cost \( C \) is the sum of the daily charge and the cost per mile driven. Since the daily charge is \\(35 and the cost per mile is \\)0.32, the function is \( C(m) = 35 + 0.32m \).

Step 3: Calculate Cost for 150 Miles

Substitute \( m = 150 \) into the cost function: \( C(150) = 35 + 0.32 \times 150 \). Calculate the total: \( C(150) = 35 + 48 = 83 \). So, the cost is \$83.

Step 4: Calculate Cost for 230 Miles

Substitute \( m = 230 \) into the cost function: \( C(230) = 35 + 0.32 \times 230 \). Calculate the total: \( C(230) = 35 + 73.6 = 108.6 \). So, the cost is \$108.60.

Step 5: Calculate Cost for 360 Miles

Substitute \( m = 360 \) into the cost function: \( C(360) = 35 + 0.32 \times 360 \). Calculate the total: \( C(360) = 35 + 115.2 = 150.2 \). So, the cost is \$150.20.

Step 6: Calculate Cost for 430 Miles

Substitute \( m = 430 \) into the cost function: \( C(430) = 35 + 0.32 \times 430 \). Calculate the total: \( C(430) = 35 + 137.6 = 172.6 \). So, the cost is \$172.60.

Key Concepts

Cost Function

A cost function is a mathematical expression used to calculate the total cost of a service or product. In the case of the Rent-Me Car Rental, the cost function represents the daily rental cost based on a fixed daily fee plus a variable fee depending on the miles driven.

The cost function is given by:

• Fixed daily charge: \(35 \) to the daily charge.
• Incorporate each driven mile, \(0.32m\), leading to the full cost function \( C(m) = 35 + 0.32m \).

Variable Definition

When working with linear functions, defining variables clearly is essential. Let's look into the specific variables for Rent-Me Car Rental's cost function. Here, the variables help us understand what each term in the mathematical expression represents.

• \( C \): Represents the total cost of renting the car for the day.
• \( d \): The fixed daily charge, which is \(\$35\) in this case.
• \( m \): Refers to the number of miles driven during the rental period. This variable determines how the mileage part of the rental fee will vary.

Defining these variables simplifies the calculation, allowing us to plug them into our cost function for any number of miles.

Arithmetic Operations

To solve the exercise using the linear function, it is important to understand arithmetic operations. These operations allow us to calculate the total cost by combining fixed and variable costs.

Here’s how we apply arithmetic operations in this exercise:

• Substitution: Replace \( m \) with the number of miles driven, like 150, 230, 360, or 430 miles in the given examples.
• Multiplication: Calculate \(0.32 \times m\), which gives the cost based on miles driven.
• Addition: Combine \(35\) (the fixed charge) and \(0.32m\), resulting in the total cost.

Using arithmetic operations step by step ensures accurate calculation of rental costs based on different distances traveled.