Problem 43 - Algebra and Trigonometry

Question

Explain what is meant by combined variation. Give an example with your explanation.

Step-by-Step Solution

Answer

Combined variation is a relationship between three or more quantities where one variable is directly or inversely proportional to other variables. An instance can be when the building cost (\(y\)) is directly proportional to the size of the house (\(x\)) and inversely proportional to the wage rate of construction workers (\(z\)). This can be written as \(y = k \cdot \frac{x}{z}\), which is the formula for combined variation.

Step 1: Explanation of Combined Variation

Combined variation is the relationship between three or more quantities. One variable can be directly or inversely proportional to other variables. The formula for combined variation is \(y = k \cdot \frac{x}{z}\), where \(k\) is the constant of variation, \(x\) is the variable that is directly proportional to \(y\), and \(z\) is the variable that is inversely proportional to \(y\).

Step 2: Providing an Example

Let's say the cost to build a house (\(y\)) depends on both the size of the house (\(x\)) and the wages of the construction workers (\(z\)). The larger the house or the higher the wages of construction workers, the higher the construction cost will be, indicating a direct variation. On the contrary, if we assume the number of workers is a fixed number and they are paid hourly, the lower the wage rate, the higher the construction cost, indicating an inverse variation.

Step 3: Formulating the Equation

The equation for the above example could be represented in the form of combined variation as \(y = k \cdot \frac{x}{z}\), where \(y\) represents the cost to build a house, \(x\) represents the size of the house, \(z\) represents the wage rate of construction workers, and \(k\) represents the constant of proportionality. If the cost of the building increases or decreases as the house's size changes or the wage rate changes, then the constant \(k\) remains the same, indicating a combined variation.

Key Concepts

Direct Variation

Direct variation describes a simple relationship where one variable changes proportionally with another. Suppose we have two variables, such as the distance driven and the amount of fuel used. These variables can be said to have a direct variation. The foundational element here is proportionality, meaning as one variable increases, the other does so at a constant rate.

Mathematically, direct variation is expressed as \( y = kx \), where:

• \( y \) is the dependent variable, changing based on \( x \).
• \( x \) is the independent variable.
• \( k \) is the constant of variation, representing the rate of change.

This equation shows that for any values \( x \) and \( y \), their ratio \( \frac{y}{x} \) always equals \( k \).

• Example: If a car uses 2 gallons of gas per 100 miles, and you drive 200 miles, you use 4 gallons. Here, the constant of variation \( k \) is 2.

Inverse Variation

Inverse variation represents a scenario where one variable increases as another decreases. This type of relationship is common in real-life situations where resources are limited.

The mathematical structure for inverse variation is \( y = \frac{k}{x} \), where:

• \( y \) is the variable that inversely depends on \( x \).
• \( x \) is the independent variable.
• \( k \) is the constant of variation, which remains the same as \( y \) and \( x \) change.

In inverse variation, the product \( yx \) equals the constant \( k \).

• Example: If the speed \( y \) of a vehicle increases, the time \( x \) it takes to travel a fixed distance decreases, assuming a constant product \( k \).

Constant of Variation

The constant of variation \( k \) is pivotal in both direct and inverse variation equations. It is the unchanging rate that links the variables, regardless of how those variables themselves change.

In direct variation \( y = kx \), the constant \( k \) represents how much \( y \) changes as \( x \) changes. For inverse variation, \( y = \frac{k}{x} \), \( k \) defines how \( y \) scales inversely with \( x \).

• Role of \( k \): Determines the slope or tendency of the relationship between the variables.
• In combined variation: \( y = k \cdot \frac{x}{z} \), \( k \) plays a central role, impacting both direct and inverse parts of the relation simultaneously.

Understanding \( k \) is essential, as it allows predictions on how changes in one variable affect another within a specific context.

Proportional Relationships

Proportional relationships underscore the essence of both direct and inverse variations. They define how variables relate to each other through a constant factor – a fundamental concept in mathematics and science.

For a relationship to be proportional, the ratio or product of the variables must remain consistent.

• Direct Variation: Proportionality manifests as a constant ratio \( \frac{y}{x} = k \).
• Inverse Variation: Here, proportionality is seen as a constant product \( y \cdot x = k \).

Proportional relationships simplify complex systems, making it easier to predict outcomes when one variable changes.

• Example in life: A recipe scaling up or down while maintaining taste and texture also follows proportional relationships by keeping ingredient ratios consistent.