Problem 11 - Algebra 2 and Trigonometry

Question

In \(6-12,\) tell whether the variables vary directly, inversely, or neither. Each day, I am awake \(a\) hours and I sleep \(s\) hours.

Step-by-Step Solution

Answer

The variables vary neither directly nor inversely.

Step 1: Understand the Relationship

We need to figure out the relationship between the two variables: the hours I am awake \(a\) and the hours I sleep \(s\). Since the total number of hours in a day is fixed at 24, we have the equation \(a + s = 24\).

Step 2: Represent with an Equation

Re-write the relationship as an equation: \(a = 24 - s\). This shows that the hours awake \(a\) depend on the hours of sleep \(s\) by subtracting \(s\) from 24.

Step 3: Analyze the Type of Variation

In direct variation, increasing one variable increases the other proportionally, stated as \(y = kx\). In inverse variation, increasing one variable decreases the other proportionally, \(y = \frac{k}{x}\). Here, \(a = 24 - s\) shows that when \(s\) increases, \(a\) decreases by the same amount, and vice versa, following a fixed sum.This represents a neither direct nor inverse variation, as the equation does not fit the forms of either direct or inverse variation.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable increases, causing the other variable to increase proportionally. This means if you have a direct variation between two quantities, as one goes up, the other goes up too. The general form of a direct variation equation is written as \( y = kx \), where \( k \) is a constant.
This is frequently called the constant of variation. For example, if you double \( x \), then \( y \) also doubles, assuming the constant \( k \) remains unchanged.

• If you think about the relationship between the hours someone works and the pay they receive at a constant rate, it's a direct variation.
• The more hours you work, the more money you make, assuming the hourly rate (constant \( k \)) doesn't change.

Inverse Variation

Inverse variation occurs when one variable increases while the other decreases proportionally. This relationship is characterized by the formula \( y = \frac{k}{x} \), where \( k \) is again the constant of variation.
In an inverse variation, as \( x \) becomes larger, \( y \) gets smaller, and vice versa, but their product is always equal to the constant \( k \).

• An example of inverse variation could be how quickly you can finish a task if more people are helping.
• The more people working on the task, the less time it typically takes to complete, provided everyone contributes equally.

Neither Variation

Sometimes relationships between two quantities don't fit the models of direct or inverse variation. This is what we call neither variation. It simply means the relationship doesn't strictly follow either pattern. In the given exercise, the relationship between awake hours \( a \) and sleep hours \( s \) is based on a fixed sum, seen in the equation \( a + s = 24 \).
Here, you can notice as \( s \) increases by an hour, \( a \) decreases by an hour, maintaining the sum of 24.
This does not fit into the equation forms of either direct (\( y = kx \)) or inverse (\( y = \frac{k}{x} \)) variations. The balance of the two variables is based on a constraint, not a proportional relationship.

• The scenario where the increase in one variable causes a decrease in another but not in a way that forms a clear reciprocal relationship, often falls into this category.

Algebraic Equation

An algebraic equation is a mathematical statement that asserts the equality of two expressions.
Equations involving variables like \( a + s = 24 \) are used to describe specific relationships between quantities.
In this form, the equation shows that the sum of hours awake and hours asleep is always equal to 24.
This equation does not imply direct or inverse variation; instead, it describes a fixed total constraint.

• Equations can help define relationships, allowing you to solve for unknowns when other information is known.
• Simple equations can reveal how adjusting one part affects the whole, as seen in how changes in \( s \) influence \( a \), but always leading to the constant total of 24.