Problem 20 - Elementary and Intermediate Algebra

Question

Express each verbal model in symbols. See Objectives 1 and 2. \(v\) varies inversely as the square of \(r\)

Step-by-Step Solution

Answer

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

Step 1: Understanding Inverse Variation

In an inverse variation model, when one quantity increases, the other quantity decreases. This is described by the relationship where the product of the two quantities is a constant.

Step 2: Identifying the Variables and Relationship

Here, we have two quantities, \(v\) and \(r\), with \(v\) varying inversely as the square of \(r\). This means \(v\) is inversely proportional to \(r^2\). To express this, we state \(v \propto \frac{1}{r^2}\), which signifies that \(v\) is inversely proportional to the square of \(r\).

Step 3: Introducing the Constant of Variation

We introduce a constant of proportionality, \(k\), to replace the proportionality sign with an equals sign. Thus, the equation becomes \(v = \frac{k}{r^2}\).

Step 4: Expressing the Final Equation

The equation \(v = \frac{k}{r^2}\) represents the verbal model 'v varies inversely as the square of r' in symbolic form, where \(k\) is a constant that remains the same as long as the variation relationship holds.

Key Concepts

Verbal Model

A verbal model is a way to describe mathematical relationships using words. It's like transforming what you know into a verbal representation before turning it into an equation. In this case, our verbal model begins with the statement: "\(v\) varies inversely as the square of \(r\)." This statement suggests that as \(r\) changes, \(v\) changes in the opposite manner because it's an inverse relationship.

When turning a verbal model into symbols, our goal is to construct an equation that captures this inverse relationship. We start by recognizing the words "varies inversely". This tells us that we need to consider the structure where the increase in one value results in the decrease of the other. The relationship is more complex here because \(v\) is related inversely to \(r^2\), not simply \(r\), which we need to keep in mind moving forward.

Proportionality

Proportionality is key to understanding inverse variation. Inverse proportionality specifically means you're looking at a relationship where one term increases as the other decreases. In an inverse relationship between \(v\) and \(r\), you see this expressed as \(v \propto \frac{1}{r^2}\).

The term \(\propto\) represents the idea of proportionality with an inverse twist. It shows that \(v\) is not directly linked to \(r\) but rather to the reciprocal of the square of \(r\). Here’s what this tells us:

• As \(r\) gets bigger, \(v\) gets smaller,
• and as \(r\) gets smaller, \(v\) gets bigger.

This is a signature trait of inverse proportionality, where the product of one quantity and its related inverse is always constant. Proportionality is a tool to better understand how certain elements are related in mathematical problems.

Constant of Variation

The constant of variation, denoted by \(k\), is what ties the relationship in inverse variation into an equation. It's the numerical value that maintains the proportional relationship across all values of \(v\) and \(r\), as long as the rule holds. By replacing the proportionality symbol with \(k\), the relationship is mathematically complete.

In our context, the constant appears in the equation as \(v = \frac{k}{r^2}\). The presence of \(k\) allows us to express \(v\) exactly when \(r\) is given.

• \(k\) remains unchanged for consistent variables.
• Any change in \(v\) dictated by a change in \(r\) will still satisfy the equation due to the consistent ratio provided by \(k\).

The constant of variation makes it easier to solve real-world problems because you can determine \(k\) from known values and then use it to predict unknown quantities. Understanding this concept is crucial when applying inverse variations to many practical scenarios.