Problem 126 - Introductory and Intermediate Algebra for College Students 4th

Question

Membership in a fitness club costs \(\$ 500\) yearly plus \(\$ 1\) per hour spent working out. A competing club charges \(\$ 440\) yearly plus \(\$ 1.75\) 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 must work out more than 80 hours in a year for the cost of membership in the first club to be less than in the second club.

Step 1: Define the problem in terms of variables and create an inequality

Let \(X\) be the number of hours one has to work out in a year. The cost of the first club is given by \(500 + X\) dollars, and the cost of the second club is given by \(440 + 1.75X\) dollars. We want the cost of the first club to be less than the cost of the second club, so we would set up the inequality as: \(500 + X < 440 + 1.75X\).

Step 2: Solve the inequality

To solve the inequality, first subtract \(X\) from both sides to get : \(500 < 440 + 0.75X \). Now subtract 440 from both sides: \(60 < 0.75X \). Finally, to solve for \(X\), divide both sides by 0.75: \(X > 80 \).

Step 3: Interpret the solution

This result means that a person must work out more than 80 hours in a year for the cost of membership at the first club to be less than at the second club. Therefore, if a person expects to work out at least 81 hours in a year, they should join the first club, because it would be more cost-effective.

Key Concepts

Variables

Variables are the foundation of any mathematical problem-solving approach. In this context, a variable is a symbol, often a letter, used to represent an unknown quantity that can change. Here, the problem asks us to find out how many hours a person needs to work out for a specific cost decision. Thus, we introduce a variable, \(X\), to represent these workout hours. By assigning \(X\) to the number of hours, we translate a real-world scenario into a mathematical language that allows further exploration and manipulation.
Understanding variables is crucial because they help us form equations and inequalities that model the problem at hand. Once we have a variable in place, it becomes much easier to form mathematical expressions that represent different scenarios.

Linear Inequality

A linear inequality is like an equation, but instead of an equals sign, it uses inequality signs such as \(<, >, \leq,\) or \(\geq\). These inequalities express a range of possible values rather than a specific solution. In our typical fitness club problem, we use an inequality to find when one club is cheaper than the other.
The expression "Costs of first club \(<\) Costs of second club" is a linear inequality. The cost of the first club is represented by \(500 + X\), and the second club by \(440 + 1.75X\). Thus, the inequality can be written as \(500 + X < 440 + 1.75X\). This setup is crucial as it provides a method to compare costs based on the variable representing workout hours.

Cost Analysis

Cost analysis involves evaluating the expenses involved to make more economical decisions. In this case, we examine two membership packages to find which is more affordable based on workout hours.
The yearly cost plus the variable hourly cost forms the total expense of each club:

• First Club: \(500 + X\)
• Second Club: \(440 + 1.75X\)

Comparing these costs requires understanding how they change with workout hours. Through solving the inequality, \(500 + X < 440 + 1.75X\), we determine at what point (number of hours) the first club becomes the cost-effective choice. Thus, cost analysis can guide our decision-making for financial optimization.

Problem Solving

Problem solving is a process involving understanding the problem, forming strategies, and implementing solutions. In math, particularly with inequalities, it starts with defining the problem using variables and setting up expressions.
1. **Define Variables**: Identify what needs to be found and assign appropriate variables.
2. **Formulate Inequalities**: Look at comparisons needed and express these as inequalities.
3. **Solve Mathematically**: Manipulate the inequalities through arithmetic operations to isolate the variable and find a solution.
4. **Interpret Results**: Consider the mathematical outcomes in the context of the original problem to draw conclusions. Here, solving \(500 + X < 440 + 1.75X\) and finding \(X > 80\) means evaluating how many workout hours tip the cost scale in favor of one club over the other, leading to an informed choice.