Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

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 \(\$ 120\) ?

Step-by-Step Solution

Answer

For \$120, you can drive 240 miles.

Step 1: Understand the Problem

The problem involves finding the total rental cost for a car. We must account for a base daily charge and a per-mile charge to write a formula for the total expense, then use this formula to find how many miles can be driven with a specific budget.

Step 1: Set Up the Formula for Total Expense

The formula for total rental expense in dollars, \(E(x)\), includes a fixed cost of \\(24 plus an additional amount for each mile driven at \\)0.40 per mile. Therefore, the formula is \[ E(x) = 24 + 0.40x \] where \(x\) represents the number of miles driven.

Step 2: Solve for Number of Miles Driven

Given a budget of \$120, set \(E(x) = 120\) and solve for \(x\). Start with the equation: \[ 24 + 0.40x = 120 \]. Subtract \(24\) from both sides to isolate the term with \(x\): \[ 0.40x = 96 \].

Step 3: Solve for x

Divide both sides by \(0.40\) to solve for \(x\): \[ x = \frac{96}{0.40} = 240 \]. Hence, \(x = 240\), meaning you can drive 240 miles for \$120.

Key Concepts

Linear Equations

Linear equations are the bedrock of algebra. They are equations that make a straight line when graphed. In our exercise, the linear equation is used to calculate the total rental expense. Here, the formula for total rental expense over a day is created. It consists of fixed and variable costs. The fixed cost represents the daily car rental charge, while the variable cost accounts for the per-mile cost driven.

The formula we work with is:

• \(E(x) = 24 + 0.40x\)

Here, \(E(x)\) symbolizes the total expense based on the miles \(x\) driven.
Understanding how to manipulate this equation helps solve real-world problems like calculating costs based on usage, depicting a linear relationship that can be crucial in budgeting and financial forecasting.

Cost Function

A cost function is a mathematical formula that helps to determine the total expenses associated with a process or activity. In the context of car rentals, the cost function helps you figure out how much it will cost to rent a car for a day, depending on mileage driven.

Our equation:

• \(E(x) = 24 + 0.40x\)

is a classic example of a cost function. Here, \(24\) is the fixed cost for car rental, and \(0.40x\) is the variable cost that changes with the distance driven. Understanding and implementing cost functions can be immensely helpful when planning and managing expenses across different scenarios. It provides clarity on how expenses will evolve as usage increases.

Problem Solving

Problem-solving skills are vital in mathematics and are mostly about understanding, setting up, and solving equations. In this exercise, we used the steps of problem-solving: understanding the problem, setting up the linear equation, and using it to find a solution.

We start by identifying what we need to find out - the number of miles you can drive under a certain budget. With our equation \(E(x) = 24 + 0.40x\), we substitute \(E(x)\) for the budget value \(120\) to find the number of miles that can be driven:

• Set \(E(x) = 120\) resulting in \(24 + 0.40x = 120\).
• Solve for \(x\) by rearranging to isolate the \(x\) term, leading to \(0.40x = 96\).
• Solve by dividing through to give \(x = 240\).

You can then confidently say you can drive 240 miles for $120, utilizing the step-by-step process of problem-solving.

Budget Constraints

Budget constraints are an important factor in almost every financial decision. They imply restrictions on spending, ensuring you do not exceed available limits. In this example, the budget constraint allows us to calculate how far one can drive without going over the \(120\) dollar limit.

By substituting our budget into the cost function:

• \(24 + 0.40x = 120\)

we managed to derive the maximum miles possible within that budget, resulting in 240 miles. Understanding budget constraints not only helps in problems involving expenses but also aids in strategic planning, allowing resources to be allocated efficiently without overstepping financial boundaries.