Problem 134 - Algebra and Trigonometry

Question

Basic Car Rental charges \(\$ 20\) a day plus \(\$ 0.10\) per mile, whereas Acme Car Rental charges \(\$ 30\) a day plus \(\$ 0.05\) per mile. How many miles must be driven to make the daily cost of a Basic Rental a better deal than an Acme Rental? (Section 1.7, Example 11)

Step-by-Step Solution

Answer

You must drive more than 200 miles for the Basic Rental to be a better deal than the Acme Rental.

Step 1: Set up the inequality

Determine the daily cost for Basic and Acme in terms of miles. For Basic, it is \(20 + 0.10m\). For Acme, it is \(30 + 0.05m\). To make Basic a better deal, its total cost should be less than Acme's. So set the inequality: \(20 + 0.10m < 30 + 0.05m\).

Step 2: Solve the inequality

In order to solve this inequality, first simplify it by subtracting \(0.05m\) from both sides and subtracting 20 from both sides. This gives us \(0.05m > 10\). By further simplifying, we get \(m > 200\).

Step 3: Interpret any result

The solution of the inequality is \(m > 200\). This means that the Basic Rental becomes a better deal than the Acme Rental when more than 200 miles are driven.

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions used to compare two quantities with a less than, greater than, or not equal to relationship. They are similar to linear equations, but instead of "equals," you use inequality signs like \(<\), \(>\), \(\leq\), or \(\geq\). Here, we compared two expressions:

• Basic Car Rental: \(20 + 0.10m\)
• Acme Car Rental: \(30 + 0.05m\)

In this problem, we aimed to find the miles, \(m\), needed for Basic to become cheaper than Acme. This requires setting up the inequality \(20 + 0.10m < 30 + 0.05m\). The challenge is manipulating and solving this inequality correctly.

Cost Comparison

Comparing costs using inequalities is a practical application in daily life. In this exercise, the cost comparison was between two car rental companies, focusing on determining which option was more economical given a certain number of miles driven.

• Basic Car Rental: Charges a base rate of \\(20 per day with an additional \\)0.10 per mile.
• Acme Car Rental: Charges a base rate of \\(30 per day with an additional \\)0.05 per mile.

Start by setting up an equation for each rental. Then, use these equations to create an inequality, which helps in comparing the costs beyond the initial pricing.

Mathematical Reasoning

Mathematical reasoning involves critical thinking and problem-solving skills to arrive at a conclusion. This exercise requires understanding and translating a word problem into a mathematical inequality. Let's break it down:

• Translate the scenario into mathematical expressions: \(20 + 0.10m\) and \(30 + 0.05m\).
• Decide what we want: Basic should be cheaper, so set \(20 + 0.10m < 30 + 0.05m\).

The reasoning process also involves breaking down the inequality step by step to understand how the problem translates from words to mathematical symbols. This is crucial for interpreting results correctly.

Inequality Solving

Solving inequalities involves steps similar to solving equations, but with one crucial difference when multiplying or dividing by a negative number (the inequality sign flips, though this is not applicable in this particular problem).Here's the process for our problem:

• Start with: \(20 + 0.10m < 30 + 0.05m\)
• Subtract \(0.05m\) from both sides: \(20 + 0.05m < 30\)
• Subtract 20 from both sides: \(0.05m > 10\)
• Divide both sides by \(0.05\): \(m > 200\)

This solution shows that driving more than 200 miles makes the Basic Rental cheaper than Acme. Keep practicing these steps, as they are foundational for mastering inequality problems.