Problem 122 - Algebra and Trigonometry

Question

In Exercises 122–133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A truck can be rented from Basic Rental for 50 dollar per day plus 0.20 dollar per mile. Continental charges 20 dollar per day plus 0.50 dollar per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal than Continental's?

Step-by-Step Solution

Answer

The number of miles a day must be more than 100 for the rental cost for Basic Rental to be a better deal than Continental's.

Step 1: Write Down the Equations

Firstly, it's important to establish the cost equations for both Basic Rental and Continental. The cost for Basic Rental (B) is given by the equation \(B = 50 + 0.20M\), where \(M\) is the number of miles driven. Similarly, the cost for Continental (C) is given by \(C = 20 + 0.50M\). The goal is to find when \(B < C\).

Step 2: Set Up the Inequality

In order to find when Basic Rental is a better deal, substitute the price equations from step 1 into inequality \(B < C\). That gives \(50 + 0.20M < 20 + 0.50M\).

Step 3: Simplify the Inequality

Now, we can simplify our inequality by subtracting \(0.20M\) and 20 from both sides, which results in \(30 < 0.30M\).

Step 4: Solve the Inequality

To isolate \(M\), divide both sides of the inequality by 0.30, yielding \(M > 100\).

Key Concepts

Linear Inequality Application

When we encounter real-world scenarios that require comparing different options based on certain restrictions, we often use linear inequalities to model and solve these situations. In our textbook example, the objective is to determine when one truck rental option becomes more cost-effective than another. This hinges on understanding that the amounts spent on the two rental options form an inequality based on the number of miles driven.

Applying a linear inequality involves translating a verbal situation into a mathematical statement. Here, if we let 'M' represent the miles driven, we are comparing the costs of Basic Rental and Continental, translating the problem into the inequality: 'Basic's cost is less than Continental's cost' expressed as the inequality: \(B < C\).

Setting Up Equations

Before we tackle the inequality, we must first articulate the given information through equations. Setting up the correct equations is essential; in our example, it requires an understanding of the cost structure of both truck rental companies. The equation for Basic Rental’s cost is \(B = 50 + 0.20M\), and for Continental, it's \(C = 20 + 0.50M\).

We then need these equations when setting up our inequality. The intuition is that as 'M' changes, which of B or C is greater will also change, and we seek the threshold where that change occurs.

Inequality Simplification

Breaking down complex expressions by inequality simplification moves us closer to the solution. The given problem already provides a vital step: transforming the inequality \(50 + 0.20M < 20 + 0.50M\) by combining like terms and isolating the variable. Simplifying the inequality by removing \(0.20M\) from each side and subtracting 20 from both sides gives us \(30 < 0.30M\).

Simplification removes the constant part of the expense, focusing on how the variable 'M' affects each option's cost, making it clearer at what point the number of miles driven makes Basic Rental less expensive than Continental.

Mathematical Modeling

The process of forming a mathematical representation of a real-world situation is known as mathematical modeling. In our example, the real-world situation is comparing the cost of two different truck rental companies. The mathematical model that emerges consists of two parts: first, the cost equations for each company, and second, the inequality that represents our objective of finding when one cost is less than the other.

When we solve the inequality \(30 < 0.30M\) and find that \(M > 100\), this means we have successfully applied a mathematical model to determine that after driving more than 100 miles, Basic Rental offers the better cost option. Mathematical models, like the one used in this example, are powerful tools in making informed decisions based on numerical analysis.