Problem 2 - Precalculus Essentials

Question

A car rental agency charges \(\$ 180\) per week plus \(\$ 0.25\) per mile to rent a car. a. Express the weekly cost to rent the car, \(f\), as a function of the number of miles driven during the week, \(x\). b. How many miles did you drive during the week if the weekly cost to rent the car was \(\$ 395 ?\)

Step-by-Step Solution

Answer

a. The cost function is \(f(x) = 180 + 0.25x\). b. If the cost was $395, the number of miles driven was 860.

Step 1: Formulate the function

The first part asks to express the weekly rental cost, \(f\), as a function of miles driven during the week, \(x\). This is given by the weekly charge of $180 plus $0.25 per mile. Hence the function will be \(f(x) = 180 + 0.25x\)

Step 2: Solve for x when the weekly cost is $395

The second part of the problem asks how many miles were driven if the cost was \$395. To find this, set \(f(x) = 395\) and solve for \(x\). The equation to solve becomes $395 = $180 + $0.25x

Step 3: Calculation and Solution

Solving the equation \(395 = 180 + 0.25x\) we start by subtracting $180 from both sides, we have \(215 = 0.25x\). Next, we divide both sides by $0.25 to solve for \(x\), giving \(x = 215 / 0.25 = 860\). Therefore, if the weekly cost to rent the car was $395, then 860 miles were driven that week.

Key Concepts

Understanding Linear Equations in Cost Functions

Linear equations are a fundamental part of mathematics and appear frequently in problems involving cost functions, like our car rental scenario. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. It can be represented in the form of \( y = mx + b \), where:

• \( y \) is the dependent variable, which changes based on the other variables.
• \( m \) is the slope of the line, indicating how much \( y \) changes with a change in \( x \).
• \( x \) is the independent variable, representing the input or cause for the change in \( y \).
• \( b \) is the y-intercept, showing the value of \( y \) when \( x \) is 0.

In the car rental example, the equation is formed as \( f(x) = 180 + 0.25x \). Here:

• \( f(x) \) is the weekly rental cost, our dependent variable.
• \( 0.25 \) is the slope, showing that for each additional mile, the cost increases by $0.25.
• \( x \) is the miles driven during the week.
• \( 180 \) is the base fee, which is the cost of renting the car before considering mileage.

Understanding how to set up and solve linear equations is crucial for calculating costs that depend on varying factors like mileage.

Calculating Weekly Rental Costs

Weekly cost calculation often involves using linear equations to find out how much you need to pay based on certain conditions. In our specific car rental example, calculating the weekly cost requires knowing both fixed and variable components. By breaking down costs this way:

• The fixed component is constant and does not change regardless of usage—in this case, the weekly base charge of \(180.
• The variable component changes based on usage, which is \)0.25 per mile driven.

When you combine both elements, you get the total weekly cost. This gives us our cost function \( f(x) = 180 + 0.25x \), allowing easy calculation by simply plugging in the number of miles driven.
For example, if you plan to drive 500 miles in a week, substitute \( x = 500 \) in the equation:
\[ f(500) = 180 + 0.25 \times 500 = 180 + 125 = 305 \]
The total weekly cost would be $305. Interpreting the components of such an equation provides insights into making financial decisions effectively.

Understanding Mileage Charges

Mileage charge is a common term in rental agreements and can significantly affect total costs. It is essential to understand how these charges accumulate over your rental period. In this context, the mileage charge of \(0.25 per mile is the variable cost affecting the overall price of renting the car.

• The mileage charge reflects the cost increase attributable to each mile driven.
• This charge means that the renter pays an additional \)0.25 for every mile beyond the baseline weekly charge.

For instance, if no miles are driven, the cost remains at the base $180. However, with each additional mile, the charge adds up to the total expenditure. Therefore, understanding how mileage affects charges can lead to more informed decision-making—whether it's for planning trips or budgeting for potential expenses.
In the second part of our problem, knowing this rate helps calculate the miles driven when given the total cost. For example:\[ 395 = 180 + 0.25x \]Solving this provides insights into how far you can travel within a certain budget.