Question

The Acme Car Rental Agency charges \(\$ 24\) a day for the rental of a car plus \(\$ 0.40\) per mile. (a) Write a formula for the total rental expense \(E(x)\) for one day, where \(x\) is the number of miles driven. (b) If you rent a car for one day, how many miles can you drive for \(\$ 120 ?\)

Short answer

(a) \( E(x) = 24 + 0.40x \); (b) You can drive 240 miles for \$120.

Step-by-step solution

Define the Variables

First, we need to define the variables involved in the problem. Let the total rental expense be denoted by \( E(x) \) and the number of miles driven be denoted by \( x \).

Establish the Cost Components for One Day

The rental fee for one day is \\(24, independent of the miles driven. Additionally, there is a cost of \\)0.40 per mile driven. Therefore, the total cost components are the fixed rental fee and the variable mileage fee.

Formulate the Rental Expense Equation

The total rental expense \( E(x) \) can be calculated by adding the day rate and the mileage cost together. The formula is given by: \[ E(x) = 24 + 0.40x \] where \( 24 \) is the fixed daily cost, and \( 0.40x \) is the variable cost depending on miles driven.

Set Up the Equation for Total Cost

For part (b), we are given the total cost \( E(x) \) as \$120 and need to find the number of miles \( x \). Set up the equation: \[ 24 + 0.40x = 120 \]

Solve for x

Subtract \( 24 \) from both sides to isolate the mileage cost: \[ 0.40x = 96 \] Next, divide both sides by \( 0.40 \) to solve for \( x \): \[ x = \frac{96}{0.40} = 240 \] Thus, you can drive 240 miles.

Key concepts

Understanding Cost Components

When calculating expenses, understanding the different cost components is essential. In our example with the Acme Car Rental Agency, there are two primary components of cost when renting a car: a flat daily fee and a fee per mile driven.

• **Fixed Costs:** These are stable, regardless of usage. For Acme, it's the daily charge of $24.
• **Variable Costs:** These vary with activity level. Here, it's $0.40 per mile driven.

Being able to break down costs into fixed and variable components helps in predicting total expenses. This breakdown enables more accurate budgeting and financial planning for individuals and businesses. Understanding these components makes equation solving, later on, more straightforward.

The Role of Variable Cost

Variable costs are dynamic; they fluctuate based on usage. In the context of car rentals, the more you drive, the higher the variable cost. This is because variable costs are directly proportional to the level of activity or service usage.

In our exercise, every mile driven adds $0.40 to the total expense. Hence, if you drive more miles, you pay more, and if you drive less, you pay less. These costs can significantly affect the total expense, making it crucial to account for them in any financial calculations involving activities that have such costs.

By understanding the variable nature of these costs, a renter can effectively manage their travel to minimize expenses.

Identifying Fixed Costs

Fixed costs remain constant, irrespective of the level of activity. For the Acme Car Rental Agency example, the $24 daily rental fee is a fixed cost. This means that whether you drive 1 mile or 100 miles, this fee remains unchanged.

The implication of fixed costs is that they are predictable and inevitable once you commit to the service for the day. They form the base of your total expenses, ensuring that no matter what, this part of the expense will be incurred. This stability is helpful for setting a baseline budget or understanding the minimum cost for a particular service.

Knowing fixed costs allows you to evaluate whether the service provides value even before factoring in variable costs.

Equation Solving for Rental Expenses

Equation solving is a methodical way to find unknown values using known relationships. With the car rental example, solving equations helps determine the maximum mileage possible under a certain budget.

The provided equation for part (b), \[ 24 + 0.40x = 120 \], represents the total cost setup where \(E(x)\) is \(120. By isolating the variable \(x\), which stands for miles, we can calculate the possible miles driven within the budget.

• Subtract \)24 from both sides to consolidate variable-dependent terms: \(0.40x = 96\).
• Divide by the variable cost rate ($0.40) to solve for \(x\): \(x = \frac{96}{0.40} = 240\).

Understanding how to manipulate and solve these equations can be applied to various scenarios where cost management and budgeting are necessary. It emphasises the practical utility of algebra beyond theoretical exercises.