Problem 3 - Precalculus: Mathematics for Calculus

Question

If the quantities \(x, y,\) and \(z\) are related by the equation \(z=3 \frac{x}{y}\), then we say that \(z\) is _______________ _______________ to \(x\) and _______________ _______________ to \(y\).

Step-by-Step Solution

Answer

directly proportional to \(x\), inversely proportional to \(y\).

Step 1: Understand the Relationship

The equation given is \( z = 3 \frac{x}{y} \). It means that the value of \( z \) depends on the values of \( x \) and \( y \). We need to determine how \( z \) changes with changes in \( x \) and \( y \).

Step 2: Analyze Relationship to x

In the equation \( z = 3 \frac{x}{y} \), if we increase \( x \) while keeping \( y \) constant, \( z \) also increases. Similarly, if we decrease \( x \), \( z \) decreases. This shows that \( z \) is directly proportional to \( x \).

Step 3: Analyze Relationship to y

In the same equation \( z = 3 \frac{x}{y} \), if \( y \) increases while \( x \) stays the same, \( z \) decreases. Conversely, if \( y \) decreases, \( z \) increases. Therefore, \( z \) is inversely proportional to \( y \).

Step 4: Fill in the Blanks

Based on the analysis of step 2 and step 3, we can conclude that \( z \) is directly proportional to \( x \) and inversely proportional to \( y \).

Key Concepts

Directly Proportional

When two quantities are directly proportional, they increase or decrease together.
In a mathematical sense, if one quantity doubles, the other will double as well, assuming the proportionality constant remains unchanged.
Let's dive into an example using the equation from the exercise:

• Given the equation: \( z = 3 \frac{x}{y} \)
• We see that as \( x \) increases (with \( y \) constant), \( z \) increases proportionally.

This means \( z \) is directly proportional to \( x \).
The constant of proportionality in this example is \( \, 3/y \, \), but what's important is that their relationship is one where change in one directly results in a similar direction of change in the other.
This is why we refer to these sorts of relationships as direct proportionality.

Inversely Proportional

Inversely proportional relationships involve one quantity increasing as the other decreases.
The product of the two quantities remains constant.
Consider again the equation:

• In \( z = 3 \frac{x}{y} \), when \( y \) increases (keeping \( x \) the same), \( z \) decreases.
• Conversely, when \( y \) decreases, \( z \) increases.

This behavior exemplifies an inverse proportionality.
Such relationships can often be visualized as one quantity shrinking as the other grows.
In practical terms, imagine if \( y \) represents the number of workers on a project and \( z \) represents the time to completion – more workers mean less time needed.

Dependency Analysis

Dependency analysis is a way to study how different variables in an equation relate to each other.
It allows us to understand the effect of changing one variable on another in a mathematical relationship.
With our equation \( z = 3 \frac{x}{y} \), we can perform a dependency analysis:

• Analyze how \( z \) depends on \( x \): Changing \( x \) directly changes \( z \).
• Analyze how \( z \) depends on \( y \): Changing \( y \) inversely impacts \( z \).

Understanding these dependencies is vital in predicting outcomes and making informed decisions based on how variables interact.
Dependency analysis can be applied beyond mathematics, helping us see the interconnectedness of real-world phenomena.