Problem 133 - 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. The toll to a bridge is 3.00 dollar. A three-month pass costs 7.50 dollar and reduces the toll to 0.50 dollar. A six-month pass costs $30 and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

Step-by-Step Solution

Answer

At least 4 times in a three-month period for the three-month pass to be the best deal.

Step 1: Set up the cost model for each option

Firstly, the cost for crossing the bridge without a pass is $3 each time as per the problem. So, if a person crosses the bridge \( x \) times, the cost would be \( 3x \) dollars. \nWhen a three-month pass is bought for $7.50, each trip thereafter costs $0.50. Therefore, the total cost for \( x \) trips with a three-month pass is \( 7.5 + 0.5x \) dollars.

Step 2: Formulate the inequality

The three-month pass becomes a better deal when its total cost is lesser than the cost of crossing without a pass. Therefore, we solve the inequality \( 7.5 + 0.5x < 3x \).

Step 3: Solve the inequality

Subtract 0.5x from both sides of the inequality to get \( 7.5 < 2.5x \). Then divide both sides by 2.5 to find \( x > 3 \). However, since you can't cross the bridge a fractional number of times, you need to cross the bridge at least 4 times for the three-month pass to be the best deal.

Key Concepts

Cost Modeling

Cost modeling is a technique used to understand the financial implications of different options. In this problem, we have three types of costs associated with crossing a bridge: no pass, a three-month pass, and a six-month pass. Each option comes with its own set of conditions and costs:

• No pass: Pay $3 each time you cross.
• Three-month pass: Costs $7.50 upfront, reduces the toll to $0.50 per crossing.
• Six-month pass: Costs $30 and allows unlimited crossings.

By establishing these models, we can compare which option is the most cost-effective based on the number of crossings. This helps to form a clear picture of costs relative to expected usage.

Word Problems

Word problems are a way to apply mathematical concepts to real-life scenarios. They require you to extract information from a given problem statement and convert it into mathematical equations or inequalities. The key steps involve:

• Understanding the problem by identifying what you need to find.
• Identifying relevant numerical data.
• Translating words into mathematical expressions.

In this exercise, the word problem provides the costs and conditions of the bridge tolls. Our goal was to determine the situations in which a three-month pass offers savings compared to paying as you go.

Inequality Solving

Inequality solving is the process of finding the range of values for which a mathematical inequality holds true. In this problem, we established the inequality: \( 7.5 + 0.5x < 3x \).
This inequality compares the costs of two crossing strategies: with the three-month pass, and without a pass.

• Subtract \(0.5x\) from both sides to simplify into \( 7.5 < 2.5x \).
• Divide both sides by 2.5 to solve for \( x \).

This tells us that the minimum number of crossings needed for the pass to be financially beneficial is 4. Through this process, inequality solving can clarify which scenarios maximize cost effectiveness.

Bridge Tolls

Bridge tolls are costs levied for the privilege of crossing a bridge. They can be structured in various ways to accommodate frequent users. In this problem:

• The standard toll is a fixed $3 per crossing.
• Pass options reduce or eliminate this cost based on upfront payment.

Understanding these toll structures is crucial for making informed decisions about cost-saving options. This kind of model helps in evaluating whether paying per use or opting for a pass benefits you more depending on how often you cross the bridge.

Pass Costs

Pass costs refer to the upfront fees paid to reduce or eliminate recurring charges. In transport models like this one, pass costs provide options for frequent travelers:

• The three-month pass ($7.50) lowers each additional trip's cost to $0.50.
• The six-month pass ($30) allows unlimited toll-free crossings.

To determine the best pass option, compare the fixed and variable costs against the number of times you plan to cross the bridge. Calculating pass costs helps to assess their value against expected usage, offering a method to save money on frequent activities.