Problem 14 - Beginning and Intermediate Algebra

Question

Write a general variation equation using \(k\) as the constant of variation. \(b\) varies directly as the cube of \(w\)

Step-by-Step Solution

Answer

The general variation equation for the given relationship, where \(b\) varies directly as the cube of \(w\), is \(b = k(w^3)\).

Step 1: Understand direct variation

Direct variation is a relationship between two variables in which one variable is a constant multiple of the other variable. In mathematical terms, the direct variation between variables b and w can be represented as: \[b = kw\]

Step 2: Determine the variation relationship

In this problem, we are given that b varies directly as the cube of w. This means that b is directly proportional to the cube of w, and it can be represented as: \[b = k(w^3)\]

Step 3: Write the general variation equation

Using the constant of variation k, the general variation equation representing this relationship is: \[b = k(w^3)\] This is the general variation equation we were asked to find.

Key Concepts

Constant of Variation

In mathematics, the "constant of variation" is an essential part of equations that express proportional relationships. When two variables, such as \( b \) and \( w \), have a direct variation relationship, the constant of variation, often represented by \( k \), signifies the consistent ratio or multiplier linking them. This constant remains unchanged when the relationship between the variables preserves its form.
Whenever you hear about direct variation, it's crucial to remember that one variable is not simply dependent on another variable—instead, it's dependent on a consistent multiplier of it.
This multiplier, or constant \( k \), enables us to translate real-world situations into mathematical equations that we can analyze or predict further. In our example, the relationship is given as \( b = k(w^3) \). Here, \( b \) varies directly as the cube of \( w \), and the variable \( k \) reflects whatever constant factor affects how \( b \) responds to changes in \( w^3 \).

• Direct variation involves a constant ratio between variables.
• The constant of variation helps us maintain this ratio.
• In mathematics, \( k \) ensures that the equation holds true as variables change.

Inverse Variation

Inverse variation describes a different kind of relationship between two variables in comparison to direct variation. In an inverse variation, one variable increases as the other decreases. Think of it as a balancing act where as one goes up, the other must come down.
Mathematically, if \( x \) and \( y \) are variables in an inverse relationship, we can represent this with the equation \( x \cdot y = k \), where \( k \) is the constant of variation. Unlike direct variation, here \( k \) is the product of the two variables and remains still while the variables fluctuate.
To better understand, imagine you are trying to distribute a fixed number of apples to a number of friends. If you increase the number of friends, each gets fewer apples. Here, the total number of apples remains the same, representing the constant \( k \), while the division among friends keeps changing based on the number. In simple terms:

• In inverse variation, increase in one variable leads to a decrease in the other.
• The constant \( k \) represents the product or 'fixed quantity' in this case.
• Such equations help model scenarios where one quantity compensates another.

Proportionality Constant

The "proportionality constant" is another name for the constant \( k \) you've encountered in equations of variation. It's the determining value that ties the relationship between the variables within either a direct or inverse variation scenario.
Understanding its role allows us to comprehend how changes in one quantity impact another. In the context of direct variation, it implies that multiplying one variable by \( k \) yields another variable. For inverse variation, it represents the unchanged product of the two variables.
When you see the term ", think of it as a guiding value that ensures variables scale or alter in a consistent manner. Let’s consider our initial example with direct variation: \( b = k(w^3) \). Here, \( k \) is the proportionality constant as it consistently applies to the cubed term, \( w^3 \), maintaining the direct relation with \( b \).

• It's essential for stabilizing relationships in variation equations.
• The proportionality constant adjusts the scale of one variable relative to another.
• In any variation, it reflects the foundational change rate.