Problem 5 - College Algebra

Question

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(r\) varies inversely as \(t\)

Step-by-Step Solution

Answer

The equation that expresses this inverse relationship between \(r\) and \(t\) with \(k\) as the constant of variation is \(rt = k\).

Step 1: Understanding Inverse Variation

In mathematics, when one variable increases and the other decreases, we call this an inverse variation or inverse proportionality. The mathematical representation of this is \(xy = k\), where \(x\) and \(y\) are the variables, and \(k\) is a constant.

Step 2: Applying the Concept to the Problem

According to our problem, \(r\) varies inversely as \(t\). We can apply the concept of inverse variation to convert it into equation form. It would look like this: \(rt = k\), where \(r\) is equivalent to \(x\), \(t\) is equivalent to \(y\), and \(k\) is the constant.

Step 3: Finalize the Equation

Our final equation, therefor, that expresses this inverse relationship is \(rt = k\). When \(r\) increase, \(t\) will decrease in order to keep \(k\) constant and vice versa.

Key Concepts

Constant of Variation

The concept of a constant of variation plays a critical role in understanding inverse variation. Imagine you have two quantities that move in opposite directions—one goes up as the other goes down—but they do so in a harmonious manner, maintaining a steady rhythm. This rhythm is characterized by a numerical value that remains unchanged regardless of how the two quantities fluctuate; we call this the constant of variation, denoted by the symbol \( k \).

When dealing with inverse variation, this constant captures the essence of the relationship between the two quantities. It is the product of the variable quantities at any point in their dance together. For instance, if we have two variables, \( x \) and \( y \), whose product is always 12, then our constant of variation, \( k \), is 12, and we can express this relationship as \( xy = 12 \). This constant \( k \) is essential for creating an equation that represents the relationship accurately.

Inverse Proportionality

Inverse proportionality is like a see-saw with perfectly balanced weights on each end; when one side goes up, the other must come down to keep the equilibrium. This relationship between two variables is called inverse proportionality or inverse variation. In the world of mathematics, it's a way to say that as one quantity increases, another quantity decreases at a rate that keeps their product constant.

In our everyday life, we can observe inverse proportionality in action. For instance, the more people there are to share a pizza, the less pizza each person gets. If we denote the number of slices per person as \( r \) and the number of people as \( t \), there's a sweet spot—a constant product—that balances the two. Mathematically, when we say \( r \) varies inversely as \( t \), it means for every increase in \( t \), \( r \) must decrease to keep the constant of variation, \( k \), the same and vice versa.

Mathematical Representation

Capturing the essence of mathematical concepts in symbolic form streamlines complex relationships into neat, manageable expressions. The mathematical representation of inverse variation is a simple yet powerful equation: \( xy = k \). Each variable in this equation plays a crucial role. Here, \( x \) and \( y \) are two variables that are inversely proportional to each other, and \( k \) is the aforementioned constant of variation, a steadfast value that does not change.

Think of this equation as a snapshot capturing the eternal balance of the inverse relationship in the universe of mathematics. When you see this equation, envision one variable rising while the other falls, always in tandem to preserve the unchanging value of \( k \). It’s a concise summary of a dynamic interaction. So for our problem at hand, when \( r \) varies inversely as \( t \), the equation \( rt = k \) succinctly encapsulates this relationship; it's the formula that conveys a constant product no matter the individual values of \( r \) or \( t \).