Problem 7 - College Algebra

Question

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(a\) is inversely proportional to the cube of \(b\)

Step-by-Step Solution

Answer

The equation that expresses the relationship is \(a = \frac{k}{{b^3}}\)

Step 1: Understand Inverse Proportionality

Given that \(a\) is inversely proportional to the cube of \(b\), we understand that as \(b\) increases, \(a\) decreases and vice versa. Also, the cube of \(b\) is referred to as \(b^3\).

Step 2: Formulate the Equation

In an inverse variation, the product of the two variables is equal to a constant (in this case \(k\)). Therefore, the relationship between \(a\) and \(b\) is expressed as \(a \cdot (b^3) = k\).

Step 3: Writing the Equation in terms of a

Typically, the equation is written in terms of one of the variables. In this case, we want to express the equation in terms of \(a\). Therefore, we algebraically manipulate to get \(a\) on its own: \(a = \frac{k}{{b^3}}\)

Key Concepts

Constant of Variation

In any variation, whether direct or inverse, the constant of variation plays a crucial role. It is essentially a fixed number that relates two variables in a proportional relationship. In the context of inverse variation, the constant of variation, represented by "\(k\)", is the product that remains unchanged when one variable changes as another inversely adjusts. For instance, in our problem, the equation \(a \cdot (b^3) = k\) uses this constant to signify that no matter how \(a\) or \(b^3\) change, their product will always equal \(k\).
Understanding this concept is important because it helps you predict how changes in one variable can influence another while still maintaining their proportional relationship with "\(k\)" as the pivot.

Proportional Relationships

A proportional relationship is a relationship between two quantities where their ratios remain constant. When dealing with inverse proportionality, as in the given exercise, one quantity increases while the other decreases in such a way that their constant product stays the same. Consider the relationship we explored: \(a = \frac{k}{b^3}\).
Here we observe that as \(b^3\) increases, \(a\) decreases so the product \(a \cdot b^3\) remains equal to the constant \(k\). Both variables are linked proportionally through \(k\), showcasing a unique attribute of inverse proportional relationships, where one factor can provide insights into the behavior of another.

Algebraic Manipulation

Algebraic manipulation involves rearranging an equation to solve for a specific variable. In this exercise, after identifying the relationship \(a \cdot (b^3) = k\), the problem requires expressing the equation in terms of the variable \(a\).
We use algebraic techniques such as division to isolate \(a\), resulting in the equation \(a = \frac{k}{b^3}\). This form is more useful when you need to calculate \(a\) for known values of \(b\), or vice versa. Mastering these manipulations not only provides flexibility in calculations but also deepens your understanding of how variables interact in mathematical relationships.