Problem 40 - Intermediate Algebra

Question

Express each verbal model in symbols. See Objectives 5 and 6. \(v\) varies inversely as the square of \(r\)

Step-by-Step Solution

Answer

\(v = \frac{k}{r^2}\), where \(k\) is a constant.

Step 1: Understand Inverse Variation

Inverse variation means that one quantity increases while the other decreases. In this problem, the variable \(v\) varies inversely as the square of \(r\). This implies that \(v\) and \(r^2\) are inversely related.

Step 2: Translate to a Formula

The general form of inverse variation is given by \(y = \frac{k}{x}\), where \(k\) is a constant. Because \(v\) varies inversely with the square of \(r\), we can write \(v = \frac{k}{r^2}\). This means that as \(r\) increases, \(v\) decreases, and vice versa.

Key Concepts

Verbal Models

Verbal models are a way of expressing mathematical relationships using words. They help to understand and describe how different quantities are related. In the context of inverse variation, a verbal model might express a situation where one quantity varies inversely with another. For instance, when we say that "\(v\) varies inversely as the square of \(r\)", it means that there is a specific relationship between \(v\) and \(r\), where increasing \(r\) leads to a decrease in \(v\).
This verbal description translates into a mathematical relationship, allowing us to understand the connection between the two variables without yet using symbols.
The key is to identify the core relationship, whether two quantities increase together, decrease together, or one increases as the other decreases. This forms the basis for building a mathematical model later on. Verbal models serve as the stepping stone for developing a deeper comprehension of mathematical concepts before any calculations are made.

Mathematical Symbols

Mathematical symbols are used to represent the relationships described by verbal models in a precise form. They provide a universal language to express mathematical ideas clearly.
In the case of inverse variation, the general expression is \(y = \frac{k}{x}\), where \(y\) and \(x\) are variables, and \(k\) is a constant. For our specific example, we substitute \(v\) for \(y\) and \(r^2\) for \(x\) to get \(v = \frac{k}{r^2}\).
This formula shows that as \(r\) increases, \(v\) decreases proportionally to the square of \(r\), since they are inversely related. Mathematical symbols bring clarity, showing how changes in one variable affect another, and making it easier to perform calculations or predictions based on these relationships.

Constant of Variation

The constant of variation, represented by \(k\), is a crucial part of any inverse variation relationship. It's an unchanging value that defines how the two variables are related.
For a relationship expressed as \(v = \frac{k}{r^2}\), the constant \(k\) tells us about the rate at which \(v\) decreases as \(r^2\) increases.
Determining the constant of variation requires knowing at least one pair of values for \(v\) and \(r\). Once \(k\) is determined, the formula can predict other values of \(v\) for any \(r\), so long as the inverse variation relationship holds.

• Constant \(k\) is like a "glue" that upholds the relationship between variables.
• It remains unchanged as both \(v\) and \(r^2\) fluctuate.

In practical problems, calculating \(k\) is often an essential step in solving the equation and making further predictions or analyses.