Problem 45 - Intermediate Algebra

Question

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ r=k t $$

Step-by-Step Solution

Answer

The variable \( r \) varies directly with \( t \) with \( k \) as the constant of variation.

Step 1: Identify the Type of Variation

The given equation is \( r = k t \). This equation represents a direct variation, where one variable is a constant multiple of another variable.

Step 2: Express the Variation in Words

In a direct variation, the relationship between the variables can be described as: The variable \( r \) varies directly with the variable \( t \), meaning \( r \) changes directly as \( t \) changes. The constant \( k \) represents the proportionality constant between these two variables.

Key Concepts

Constant of Variation

The constant of variation, often denoted by the letter \( k \), is a crucial element in understanding direct variation equations. In a direct variation, one variable is a simple multiple of another, and this multiplier is the constant of variation.
For example, in the equation \( r = k t \), \( k \) determines how many times \( t \) counts towards the value of \( r \).
If \( k \) is 5, then \( r \) will be 5 times whatever \( t \) is. This constant ensures that the relationship between the two variables is consistent.

• Notice that \( k \) does not change as long as the conditions of the variation stay consistent.
• It provides a constant ratio between corresponding values of the variables involved.

The constant of variation tells us how much one variable changes in relation to another in a direct variation.
This concept is essential because it helps to easily predict one variable when knowing the other.

Proportionality Constant

The proportionality constant, also known as the constant of proportionality, is essentially the same as the constant of variation. In a direct variation, it links two variables directly, meaning as one increases or decreases, the other does so proportionally with respect to this constant.
The term comes into play when expressing real-world relationships mathematically. In the equation \( r = k t \), \( k \) is the proportionality constant. This indicates that \( r \) is directly proportional to \( t \).

• When \( k \) is positive, both variables increase together.
• If \( k \) is negative, they change in opposite directions.
• The size of \( k \) affects how steep or flat the rate of change is between the variables.

This constant can be thought of as the "rate" at which the dependent variable changes per unit increase in the independent variable.

Mathematical Equation

A mathematical equation is a statement asserting the equality between two expressions, often involving variables and constants. In direct variation, equations such as \( r = k t \) elegantly illustrate this relationship. Here, you see that \( r \) depends directly on \( t \), scaled by the constant \( k \).
This type of equation shows how input values (like \( t \)) can be transformed to output values (\( r \)) using the constant \( k \).
Understanding these equations:

• They offer a way to express relationships in a precise, numerical form.
• Help in predicting values of one variable when you know the value of another.
• Are fundamental in analyzing real-world scenarios like speed (distance = rate × time).

Learning how to read and interpret these equations is fundamental to grasping the concept of direct variation and using it effectively in problem-solving.