Problem 3 - Elementary Algebra

Question

Explain why the relationship of the number of bags of leaves per hour that are raked, \(x\), and the hours it takes to rake a yard, \(y\), is an inverse variation.

Step-by-Step Solution

Answer

The relationship is an inverse variation because as the number of bags raked per hour increases, the total hours to rake the yard decreases proportionally, fitting the form \( y = \frac{k}{x} \).

Step 1: Understand Inverse Variation

In inverse variation, if one variable increases, the other variable decreases proportionally. The relationship can be expressed by the equation: \[ y = \frac{k}{x} \]where \( k \) is a constant.

Step 2: Identify the Variables

Here, the number of bags of leaves per hour is represented by \( x \) and the hours it takes to rake the yard is represented by \( y \).

Step 3: Formulate the Relationship

Given the inverse variation, the relationship between \( x \) and \( y \) can be represented as: \[ y = \frac{k}{x} \]for some constant \( k \).

Step 4: Interpret the Practical Meaning

If more bags (\( x \)) are raked per hour, the total hours (\( y \)) required to rake the yard decreases. Conversely, if fewer bags (\( x \)) are raked per hour, the total hours (\( y \)) increase. This indicates an inverse relationship where \( y \) is inversely proportional to \( x \).

Key Concepts

inverse proportionality

In an inverse proportionality relationship, when one variable increases, the other variable decreases. This type of relationship is described mathematically by the equation \( y = \frac{k}{x} \), where \( k \) is a constant. Essentially, as one factor grows, the other shrinks proportionally.

In the context of our exercise, let's consider the number of bags of leaves raked per hour, denoted as \( x \), and the hours taken to rake a yard, denoted as \( y \). If more bags of leaves (\( x \)) are raked per hour, it takes fewer hours (\( y \)) to finish the task, and vice versa. This signifies that \( x \) and \( y \) have an inverse proportional relationship. This specific kind of relationship falls under what we refer to as inverse variation.

relationship between variables

Understanding the relationship between variables is key in grasping inverse variation. In our problem, the variables are the number of bags of leaves raked per hour (\( x \)) and the total hours taken to rake the yard (\( y \)). These variables are inversely related.

To put it simply, \( x \) and \( y \) interact such that as one goes up, the other goes down. This can be crucial in practical scenarios where optimizing time and effort is essential.

For example, if a team increases their efficiency by filling more bags of leaves per hour, they’ll finish the job quicker. On the other hand, if they’re less efficient, taking fewer bags per hour, they’ll need more hours to complete the yard work. This practical understanding can then be expressed and quantified through our formula: \( y = \frac{k}{x} \).

constant of variation

The constant of variation, \( k \), is a pivotal element in the equation of inverse proportionality. This constant helps establish the specific relationship between \( x \) and \( y \).

In the formula \( y = \frac{k}{x} \), \( k \) remains unchanged irrespective of the values that \( x \) and \( y \) take on. It serves as the multiplier that balances the equation.

Let’s break it down in context: Suppose you are raking leaves and your efficiency allows you to fill 10 bags of leaves per hour. If it takes 2 hours to rake the entire yard, the constant of variation (\( k \)) would be \( 20 \) because \( y = \frac{k}{x} \) or \( 2 = \frac{20}{10} \). Here, \( k = 20 \) encapsulates the particular relationship between your effort and time, which remains constant.

Having a constant \( k \) helps predict and understand how changes in one variable affect the other. For any modifications in \( x \), you can find the corresponding \( y \) accurately, ensuring a consistent proportional dynamic between the variables.