Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: The cost of renting a car is $$\$ 45 /$$ wk plus $$\$ 0.25 / \mathrm{mi}$$ traveled during that week. An equation to represent the cost would be \(y=45+.25 x,\) where \(x\) is the number of miles traveled. What is your cost if you travel 50 mi?

Step-by-Step Solution

Answer

The cost for traveling 50 miles is $57.50.

Step 1: Identify Variables and Equation

In the equation \( y = 45 + 0.25x \), \( x \) represents the number of miles traveled, and \( y \) represents the total cost of renting the car. We need to find the cost for \( x = 50 \) miles.

Step 2: Substitute the Known Value

Substitute \( x = 50 \) into the equation. The equation becomes: \( y = 45 + 0.25 \times 50 \).

Step 3: Calculate the Mileage Cost

First, calculate \( 0.25 \times 50 \), which equals \( 12.5 \). This is the cost for the miles traveled.

Step 4: Add the Weekly Base Cost

Add the weekly base cost to the cost of the miles traveled: \( y = 45 + 12.5 \).

Step 5: Compute the Total Cost

Calculate \( 45 + 12.5 \) to find the total cost. The total cost is \( y = 57.5 \).

Key Concepts

Cost Function

The cost function is a mathematical expression used to describe the total cost associated with renting a car in this scenario. It helps in understanding how the total cost is computed considering both fixed and variable components. The fixed component here is the rental cost per week, which is a constant value of \(45\) dollars. This means, regardless of the miles driven, this base cost remains constant.
The variable component is the cost per mile traveled, which is \(0.25\) dollars per mile. This part of the function increases as the number of miles traveled increases. Thus, the total cost function is given by \(y = 45 + 0.25x\), where \(y\) represents the total cost and \(x\) is the number of miles traveled. The function clearly outlines how costs escalate with distance traveled.

Variables Identification

In the equation \(y = 45 + 0.25x\), the identification of variables is crucial for understanding and solving the problem. Each part of this equation has a specific role. Here, \(x\) represents the number of miles traveled in a week. This is the variable component of the cost function as it changes depending on miles.
On the other hand, \(y\) denotes the total cost of renting the car, which is what we need to determine. Recognizing these variables and their roles helps set the stage for further steps like substitution and calculations, providing a clear path to the solution.

Substitution Method

The substitution method is a straightforward technique to solve equations involving variables. By knowing the number of miles traveled in our example, which is 50 miles, we can replace the variable \(x\) with 50 in the equation.
This substitution transforms the abstract equation into a calculable form: \(y = 45 + 0.25 \times 50\). By substituting specific values into the variable, it simplifies the process to find \(y\), the total cost. This method is essential in breaking down the equation step by step, making it easier to calculate the final solution.

Cost Calculation

Cost calculation involves two primary operations: multiplication and addition. Once substitution is done, we start multiplying to find the cost for the number of miles traveled. Here, multiply \(0.25\) by the 50 miles, which gives us \(12.5\). This product represents the variable cost based on mileage.
After determining this part, we add it to the fixed weekly cost of \(45\) dollars. The total cost computation then becomes \(y = 45 + 12.5\). Finally, we add these two values to find the total cost of renting the car for a week, which amounts to \(y = 57.5\). Thus, the total cost for traveling 50 miles is \(57.5\) dollars, combining both fixed and variable costs.