Problem 14 - Elementary and Intermediate Algebra

Question

Tell whether each relationship suggests direct or inverse variation. The cost to remodel a house and the number of square feet to be added.

Step-by-Step Solution

Answer

The relationship suggests direct variation.

Step 1: Define Direct Variation

Direct variation occurs when two variables increase or decrease proportionally. That means if one variable doubles, the other one also doubles.

Step 2: Define Inverse Variation

Inverse variation occurs when one variable increases while the other decreases. If one variable is halved, the other one is doubled in inverse variation.

Step 3: Analyze the Relationship

The cost to remodel a house typically increases as more square feet are added. Hence, as the number of square feet increases, the cost will likely increase proportionally.

Step 4: Determine the Variation Type

Since the cost and number of square feet to be added both increase together, this suggests a direct variation relationship.

Key Concepts

Inverse Variation

Inverse variation describes a situation where one quantity increases as another quantity decreases. Think of it like a seesaw; as one side goes up, the other must go down to balance it. Mathematically, we relate inverse variation by the formula \( xy = k \), where \( x \) and \( y \) are the variables, and \( k \) is a constant. This means if you multiply one variable by a constant, the other must adjust so their product remains the same.

For example, consider two variables: speed and travel time. If a person travels at higher speeds (increased variable), the time taken to reach a destination decreases (decreased variable), provided the distance remains constant. This is a classic example of inverse variation.

• Key Idea: One goes up, the other goes down.
• Formula: \( xy = k \).
• Example: Speed and time for a fixed distance.

Proportional Relationship

In mathematics, a proportional relationship describes a relationship between two variables where their ratio is constant. If you have ever adjusted a recipe while cooking, you were dealing with a proportional relationship. It means that an increase in one variable leads to a direct increase in another by the same factor. The formula used to express this is \( y = kx \), where \( k \) is the constant of proportionality.

Let's consider the cost of apples. If one apple costs $2, then the cost of \( x \) apples is \( 2x \). Here, the cost remains in a consistent ratio to the number of apples, clearly demonstrating a proportional relationship.

• Key Idea: Balance through constant ratio.
• Formula: \( y = kx \).
• Example: Cost of items per unit price.

Algebraic Relationships

Algebraic relationships encompass various types of connections between variables, using operations such as addition, subtraction, multiplication, and division. These include both direct and inverse variations and many others. An algebraic relationship can help you solve equations that describe real-world situations.

Consider the algebraic relationship used with the direct variation in our original problem, \( y = kx \), which indicates that one variable is a constant multiple of another. Understanding these relationships is fundamental when modeling situations mathematically. They form the basis for creating formulas that describe everything from simple profits, based on cost and sales, to complex engineering problems.

• Key Idea: Uses mathematical operations to relate variables.
• Types include: Direct and inverse variation.
• Applications: Wide-ranging from simple to complex problems.