Problem 57 - College Algebra

Question

For the following exercises, use this scenario: The cost of renting a car is \(\$ 45 / \mathrm{wk}\) plus \(\$ 0.25 / \mathrm{mi}\) traveled during that week. An equation to represent the cost would be \(y=45+0.25 x\), where \(x\) is the number of miles traveled. What is your cost if you travel \(50 \mathrm{mi}\) ?

Step-by-Step Solution

Answer

The cost is \$57.50 if you travel 50 miles.

Step 1: Understand the Equation

The equation for the cost to rent a car is given by \( y = 45 + 0.25x \), where:- \( y \) is the total cost in dollars.- \( x \) is the number of miles traveled.

Step 2: Substitute Miles into the Equation

Substitute \( x = 50 \) into the equation to find \( y \).This means replacing \( x \) with 50 in the equation: \( y = 45 + 0.25(50) \).

Step 3: Calculate the Cost for Miles Traveled

Calculate \( 0.25 \times 50 \) to find out the cost of traveling 50 miles.This results in \( 0.25 \times 50 = 12.5 \).

Step 4: Add the Weekly Cost

Add the weekly rental cost of \(45\) to the cost of miles traveled.So, \( y = 45 + 12.5 \).

Step 5: Calculate the Total Cost

Simplify the addition to find the total cost: \( y = 45 + 12.5 = 57.5 \).

Key Concepts

Cost Function

In the context of linear equations, a cost function describes the total cost associated with an activity or operation. Here, the cost function is represented by the equation \( y = 45 + 0.25x \). This equation is composed of two parts:

• A fixed cost of \( \\(45 \). This charge is constant, meaning it doesn't change irrespective of the number of miles traveled. This fee applies as a flat rate for renting the car for a whole week.
• A variable cost of \( \\)0.25 \) per mile, denoted as \( 0.25x \). This part of the equation increases the total cost depending on the number of miles you drive during that week. The more miles you travel, the higher this component becomes due to the multiplication by the number of miles \( x \).

For example, if you rent a car and don't travel any miles, you only pay the flat \( \$45 \) fee. However, more mileage leads to a higher overall cost.

Substitution Method

The substitution method is a powerful tool for solving equations, particularly when determining unknown values. In this scenario, the number of miles traveled, \( x \), affects the total cost \( y \). To find the cost for a specific number of miles, we use substitution.

To use this method:

• Identify the specific value for \( x \) you want to evaluate. Here, it's \( x = 50 \), meaning 50 miles will be traveled.
• Replace every instance of the variable \( x \) in the equation with the chosen value. For this equation, substitute 50 into \( y = 45 + 0.25x \) so it becomes \( y = 45 + 0.25(50) \).

By substituting the variable with a specific number, you can simplify the equation and move towards calculating the total cost efficiently.

Step-by-Step Calculation

Following a detailed, step-by-step approach to solve problems like these ensures clarity and accuracy in your calculations. Let's go through calculating the total cost for traveling 50 miles using the given equation.

• Step 1: Plug in the value for miles. Since \( x = 50 \), we replace \( x \) in the equation \( y = 45 + 0.25x \) with 50, resulting in \( y = 45 + 0.25(50) \).
• Step 2: Multiply the miles cost. Calculate \( 0.25 \times 50 \), giving \( 12.5 \). This value represents the total cost of the 50 miles traveled.
• Step 3: Add the weekly rental cost. The accumulated cost from driving is added to the fixed weekly cost of \( \$45 \), so the equation becomes \( y = 45 + 12.5 \).
• Step 4: Finalize the total. Perform the final addition, \( 45 + 12.5 \), to get \( 57.5 \).

Following these steps ensures that you methodically find the cost for a given scenario, in this case, traveling 50 miles in a rental car.