Problem 51 - Algebra A Combined Function

Question

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies inversely as \(x^{3}\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{k}{x^3}\).

Step 1: Understand Inverse Variation

In inverse variation, when one variable increases, the other decreases. The two variables are inversely proportional, meaning their product is a constant. Mathematically, if a variable \(y\) is inversely proportional to another variable \(x\), it can be expressed as \(y \cdot x = k\) or \(y = \frac{k}{x}\).

Step 2: Identify the Given Relationship

The problem states that \(y\) varies inversely as \(x^3\). This means that the product of \(y\) and \(x^3\) is a constant, denoted by \(k\). Thus, \(y \cdot x^3 = k\).

Step 3: Express the Equation Using the Constant of Proportionality

Rearrange the equation from Step 2 to solve for \(y\). Divide both sides by \(x^3\) to get \(y\) by itself: \(y = \frac{k}{x^3}\). This equation represents the inverse variation of \(y\) with \(x^3\), with \(k\) as the constant of proportionality.

Key Concepts

Constant of Proportionality

In mathematics, the constant of proportionality is a key element in relationships involving proportional variables. Think of it as the glue that binds the interactions between two variables, indicating how one variable changes in relation to another.

For simple direct variations, if two variables, say \(a\) and \(b\), are directly proportional, this can be represented as \(a = kb\), where \(k\) is the constant of proportionality. Here, \(k\) tells us how much \(a\) alters for a unit change in \(b\).

• In direct variation, the value of \(k\) is always positive.
• Each variable increases or decreases together by the same factor \(k\).

When we deal with inverse variation, the concept remains similar but with a twist. If \(y\) varies inversely with \(x^3\), we express it as \(y \cdot x^3 = k\). In this case, \(k\) is the constant that ensures the inverse relation holds. This constant helps maintain the balance despite one variable's increase causing the other's decrease.

• For inverse relationships, as one variable increases, the other decreases proportionally.
• \(k\) remains constant, ensuring the product of the varying quantities is always the same.

Inverse Proportionality

Inverse proportionality is a fascinating concept that flips the common understanding of variable relationships. In contrast to direct proportionality, where both variables rise and fall together, inverse proportionality is akin to a seesaw.

Here, the increase of one variable results in a decrease in the other. This relationship is articulated through the equation: \(y \cdot x^3 = k\), where \(k\) is the constant of proportionality.

• In an inverse proportion, the product of the two variables is a fixed constant.
• If one variable becomes twice larger, the other must become half smaller, and so forth.

Understanding this concept is crucial not only for mathematical equations but also for real-world phenomena like gravitational force, where the force decreases as the distance increases, staying true to an inverse relationship. By capturing the invariant nature of these interactions, inverse proportionality provides insight into behaviors where mutual balancing rules apply.

Algebraic Equations

Algebraic equations are the backbone of mathematical expressions. They help us organize and express relationship dynamics between variables using constants and operations. An equation like \(y = \frac{k}{x^3}\) succinctly represents an inverse variation relationship by relating \(y\), \(x\), and \(k\).

Here's how algebraic equations typically work:

• They contain variables (like \(x\) and \(y\)) and constants (e.g., \(k\)).
• Operations such as addition, subtraction, multiplication, and division are used.

Equations enable us to solve unknowns by referring known quantities. By rearranging such expressions, we can determine the value of one variable in terms of others, much like solving a puzzle.
An algebraic equation in inverse variation allows us to isolate a variable, such as finding \(y\) from \(\frac{k}{x^3}\). This makes these equations invaluable tools for prediction and analysis, particularly when studying how changes in influencing factors affect outcomes.