Problem 19 - Elementary and Intermediate Algebra

Question

Express each verbal model in symbols. See Objectives 1 and 2. \(z\) varies inversely as the cube of \(t\)

Step-by-Step Solution

Answer

The symbolic representation is \(z = \frac{k}{t^3}\), where \(k\) is a constant.

Step 1: Understanding Direct and Inverse Relationships

When a variable varies directly with another, it means they change proportionally; if one increases, the other increases. In contrast, if a variable varies inversely with another, when one variable increases, the other decreases.

Step 2: Translating Inverse Variation into a Formula

In an inverse variation, we use the formula \(z \times t^3 = k\), where \(k\) is a constant. For inverse variation of \(z\) with the cube of \(t\), it means \(z\) is proportional to \(\frac{1}{t^3}\).

Step 3: Final Symbolic Representation

By simplifying using inverse variation, we express \(z\) as \(z = \frac{k}{t^3}\), where \(k\) is a constant of proportionality. This formula represents the relationship where \(z\) varies inversely as the cube of \(t\).

Key Concepts

Direct Variation

Direct variation is a foundational concept in mathematics that describes a relationship between two variables where one variable changes in direct proportion to the other. That means if one variable doubles, the other also doubles, assuming the proportionality constant remains the same.

When we talk about direct variation, we refer to the formula: \[ y = kx \] Here, "\(y\) varies directly as \(x\)" implies that \(y\) is directly proportional to \(x\). The constant \(k\) in this formula is known as the "proportionality constant," which we'll explore further in the next section.

• In simple terms, a larger \(x\) leads to a larger \(y\) and vice versa, consistent with the multiplication operation needed to calculate \(y\).
• Examples of direct variation include distances traveled at a constant speed (distance varies directly as time) and the total cost of items when the price per item is fixed.

Understanding direct variation helps when differentiating it from inverse variation, where variables react oppositely. In the problem you worked on, it involved inverse variation, meaning the variables impacted each other in the opposite way.

Proportionality Constant

The proportionality constant, represented as \(k\), is a vital part of both direct and inverse variations. In these relationships, \(k\) is a fixed value that relates the two variables involved.

For direct variation, it's the factor that scales up or scales down to maintain the proportion. In the equation \( y = kx \), \(k\) determines how the change in \(x\) translates to a change in \(y\). For inverse variation, the role of \(k\) is equally important but functions differently:

• In inverse variation, as seen in \( z = \frac{k}{t^3} \), \(k\) maintains the inverse proportionality by determining how a change in \(t\)'s cube affects \(z\).
• Since \(k\) is constant, it doesn't change if the variables change, keeping the relationship stable and predictable.

Having a solid grasp of the proportionality constant enhances your ability to handle equations describing real-world scenarios. It lets you predict how altering one variable impacts the other consistently.

Mathematical Modeling

Mathematical modeling involves creating mathematical formulas to represent and solve real-world problems. In the case of variation problems, mathematical models help describe how two variables are related, either directly or inversely, and predict future outcomes based on these relationships.

When dealing with inverse variation, the main goal is to construct an equation like \( z = \frac{k}{t^3} \) that accurately reflects the interaction between variables. Here are the steps typically involved:

• Identify the type of variation involved, whether it's direct or inverse. This problem involves inverse variation, which affects how we construct our model—\(z\) is proportional to the inverse of the cube of \(t\).
• Determine the proportionality constant \(k\). Often, this requires additional data or conditions to calculate \(k\).
• Use the mathematical model to analyze how changes in one variable impact the other. Potentially examine scenarios where the model applies in real life, such as physics problems or economic models.

Mathematical modeling thus serves as a bridge, translating real-world phenomena into an understandable mathematical language. By seeing how variables vary, we gain predictive power and insight into the systems we're studying.