Problem 48 - Intermediate Algebra

Question

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ d=\frac{k}{W^{4}} $$

Step-by-Step Solution

Answer

\( d \) varies inversely as \( W \) to the fourth power.

Step 1: Identify the Variation Type

The given equation is \( d = \frac{k}{W^{4}} \). This resembles an inverse variation equation where the dependent variable is inversely proportional to the independent variable raised to a power.

Step 2: Write the Variation Model in Words

Based on the equation \( d = \frac{k}{W^{4}} \), we can say that \( d \) varies inversely as \( W \) raised to the fourth power. In words, this means that \( d \) decreases as \( W \) increases, assuming \( k \) is positive.

Key Concepts

Constant of Variation

In any mathematical model involving variation, the "constant of variation" or "proportionality constant" plays a crucial role. In the equation \( d = \frac{k}{W^4} \), \( k \) is the constant of variation. This constant signifies a fixed value that relates the two variables, \( d \) and \( W \), and determines the behavior of the relationship between them.

For inverse variation models like this one, the constant of variation influences how quickly or slowly the dependent variable changes in response to the independent variable. Even though the variables' behavior might seem dynamic, \( k \) remains unchanged unless explicitly altered.

Key aspects to remember about the constant of variation in inverse relationships include:

• It serves as a multiplier that scales the ratio of variation.
• Its value must remain consistent to maintain the relationship defined by the equation.
• In this context, the larger the \( k \), the greater the magnitude of \( d \), given the same value of \( W \).

Dependent Variable

In a variation equation like \( d = \frac{k}{W^4} \), the dependent variable is the one whose value depends on the change in another variable. Here, \( d \) is the dependent variable. This means the value of \( d \) is determined by the value assigned to \( W \).

Understanding the behavior of the dependent variable is crucial because:

• It helps us see how changes in the independent variable affect it.
• It provides insights into the nature of the relationship when plotted, e.g., seeing how \( d \) gets smaller as \( W \) increases.

The concept of dependency denotes that \( d \) cannot change unless there is a change in \( W \). It's important for students to visualize how \( d \) reacts as \( W \) changes when analyzing such equations.

Independent Variable

In the given equation, \( W \) is the independent variable. This is the variable that stands alone and isn't changed by the other variables you are trying to measure. Instead, it affects the dependent variable, \( d \).

In many scientific experiments and mathematical equations, knowing which variable is independent is vital because:

• It helps set the input values against which other variables will react or change.
• It can be manipulated to observe the resulting effects on the dependent variable.

For example, in \( d = \frac{k}{W^4} \), manipulating \( W \) will lead to predictable changes in \( d \). As the term independent suggests, it serves as the starting point or the initial value that prompts change in \( d \). This helps in controlled analysis and understanding of how the equation behaves as \( W \) varies.

Inverse Proportionality

Inverse proportionality refers to the type of relationship where one variable increases as the other decreases. This is exactly what is happening in the equation \( d = \frac{k}{W^4} \). Here, \( d \) is inversely proportional to \( W^4 \). This means that as \( W \) gets larger, \( d \) gets smaller, assuming all other factors remain constant.

Key points to understanding inverse proportionality include:

• The product of the two variables is a constant, i.e., \( d \times W^4 = k \).
• This relationship models scenarios where increases in one dimension lead to decreases in another, like speed and time for a constant distance.

Inverse proportionality is a fundamental concept seen in many real-life applications, such as physics and economics. Getting a grasp on how variables interact through models like these is essential for analyzing complex systems.