Problem 39 - Intermediate Algebra

Question

Express each verbal model in symbols. See Objectives 5 and 6. \(z\) varies inversely as the cube of \(t\)

Step-by-Step Solution

Answer

The equation is \( z = \frac{k}{t^3} \).

Step 1: Understand Inverse Variation

Inverse variation means that when one quantity increases, the other decreases. Mathematically, it can be represented as \( x \cdot y = k \), where \( k \) is a constant. Here, as one value goes up, the other goes down, and vice versa.

Step 2: Identify the Quantities

In the given problem, \( z \) varies inversely with the cube of \( t \). So, the relationship is between \( z \) and \( t^3 \).

Step 3: Write the Inverse Variation Formula

Since \( z \) varies inversely with \( t^3 \), we write that relationship as: \( z \cdot t^3 = k \), where \( k \) is the constant of variation.

Step 4: Solve for \( z \)

To explicitly express \( z \) in terms of \( t \), rearrange the equation: \( z = \frac{k}{t^3} \). This equation represents that \( z \) decreases as \( t \) increases, maintaining the inverse variation.

Key Concepts

Constant of Variation

In inverse variation, as one variable increases, the other decreases in such a way that their product remains constant. This constant value is what we refer to as the **Constant of Variation**. It's a key component that holds the relationship between changing variables together. Understanding this constant helps in determining how one value adjusts when another changes.

For the specific case of inverse variation of \( z \) and the cube of \( t \), it is represented by the equation:

• \( z \cdot t^3 = k \)

where \( k \) is the constant of variation. This equation implies that for any values of \( z \) and\( t \), multiplying \( z \) by the cube of \( t \) will always equal \( k \). Regardless of the numerical values, \( k \) remains unchanged, providing a consistent relationship between the variables.

Mathematical Symbols

Mathematical symbols play a crucial role in efficiently conveying complex ideas using minimal language. They serve as a universal language in mathematics that transcends linguistic barriers.

In the context of inverse variation, symbols like \( \cdot \) signify multiplication, and \( ^3 \) indicates that a variable is taken to the power of three, known as cubing. These symbols transform lengthy sentences into concise mathematical expressions. By using symbols:

• The relationship \( z \) varies inversely as \( t \) is quickly captured as \( z \cdot t^3 = k \).
• The rearrangement to express \( z \) is simply \( z = \frac{k}{t^3} \).

These symbolic representations make it easier to manipulate equations and visualize relationships between variables clearly. Their concise nature helps in focusing on the mathematical processes rather than the verbal description.

Cube of a Variable

The term "cube of a variable" refers to raising a variable to the third power. When you see \( t^3 \), it means you multiply \( t \) by itself three times. So, \( t^3 = t \times t \times t \). Cubing has a significant effect on the variable because it grows the value exponentially.

In an inverse variation scenario, using the cube of a variable amplifies the variation between the two related quantities. For example:

• If \( t = 2 \), then \( t^3 = 8 \).
• If \( t = 3 \), then \( t^3 = 27 \).

This sharp rise in value due to cubing means changes in \( t \) significantly impact \( z \). As \( t \) increases, \( t^3 \) grows rapidly, causing \( z \) to decrease even more sharply if they maintain their inverse relationship. This relationship, captured by cubing, helps in understanding complex real-world phenomena where such intense variations occur.