Problem 126 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Membership in a fitness club costs 500 dollars yearly plus 1 dollars per hour spent working out. A competing club charges 440 dollars yearly plus 1.75 dollars per hour for use of their equipment. How many hours must a person work out yearly to make membership in the first club cheaper than membership in the second club?

Step-by-Step Solution

Answer

A person needs to work out more than 120 hours in a year for the membership in the first club to be cheaper than the second one.

Step 1: Set Up the Equations

Let \(x\) represent the number of hours worked out in a year. Then, the cost of the first club can be represented as the equation \(y_1 = 500 + x\), where \(y_1\) is the total cost of the first club for the year. Similarly, the cost of the second club can be represented as the equation \(y_2 = 440 + 1.75x\), where \(y_2\) is the total cost of the second club for the year.

Step 2: Find the Point of Intersection

The point of intersection of the two lines represents the number of hours at which the cost of both clubs are the same. To find this, we set the two equations equal to each other and solve for \(x\): \(500 + x = 440 + 1.75x\). Thus, \(x = (500 - 440) / (1.75 - 1) = 120\).

Step 3: Interpret the Result

The result \(x = 120\) means that for any number of hours greater than 120, the cost of the first club will be less than the cost of the second club. Thus, a person must workout more than 120 hours in a year to make the first club cost less than the second.

Key Concepts

Cost Comparison in Choosing Fitness Clubs

When deciding between two fitness clubs, cost comparison is a fundamental concept. Simply put, it involves determining which option will be cheaper based on given conditions. Imagine you have two memberships, each with its own baseline yearly fee and cost per hour for working out. To find out when one membership becomes more cost-effective than the other, you need to look at the total costs over a certain number of workout hours.
This situation involves:

• A fixed annual membership fee for each club.
• An additional cost for each hour spent at the club.

By calculating the total yearly cost for a specific number of hours, you can compare the expenses and make an informed decision about which club offers a better deal based on your workout habits.

Algebraic Modeling with Equations

To effectively compare costs, algebraic modeling comes into play. This involves using algebra to create mathematical expressions that represent real-world situations.
Let's break it down:

• The first club's cost model is represented by the equation: \( y_1 = 500 + x \)
• The second club's cost model uses: \( y_2 = 440 + 1.75x \)

Here, \(x\) is an algebraic symbol representing the number of hours worked out in a year. These equations allow us to model the cost relationship for each club based on workout hours. By solving these equations, we can find the point at which the costs equal each other, helping to determine which option is more economical for a given scenario.

Understanding Inequalities for Decision Making

Inequalities are essential tools for decision-making processes such as choosing between fitness clubs. In this context, they help us find when one club becomes cheaper than the other.
To understand this:

• Set the algebraic models of both clubs equal to find the crossover point: \( 500 + x = 440 + 1.75x \)
• Solve for \(x\) to determine when both clubs cost the same: \( x = 120 \).

This solution tells us that working out more than 120 hours a year makes the first club less expensive.
Using inequalities such as \( x > 120 \), you can analyze how different workout time commitments affect your total cost.
Ultimately, understanding and applying inequalities allow you to make optimal financial decisions by identifying thresholds in cost comparisons.