Problem 50 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression is: "\( L \) varies jointly with \( m \) and \( n \)."

Step 1: Identify the Variables and the Model

The equation given is \( L = k m n \). In this equation, \( L \) is the dependent variable, while \( m \) and \( n \) are independent variables. The constant of variation is \( k \). This is a model of joint variation.

Step 2: Define Joint Variation

Joint variation means that a variable is directly dependent on the product of two or more other variables. Here, \( L \) varies jointly with \( m \) and \( n \).

Step 3: Express in Words

The model \( L = k m n \) expressed in words is: \( L \) varies jointly with \( m \) and \( n \). The joint variability is determined by the constant of variation, \( k \).

Key Concepts

Constant of Variation

In the context of joint variation, the constant of variation is a pivotal component. It links the relationship between the dependent and independent variables. For the equation \( L = k m n \), the symbol \( k \) represents the constant of variation. It serves as a proportionality constant that helps define the scale of the relationship. This means that for each specific value of \( k \), the product of the independent variables, \( m \) and \( n \), will determine the value of \( L \). In simpler terms:

• If \( k \) is larger, \( L \) responds more dramatically to changes in \( m \) and \( n \).
• If \( k \) is smaller, \( L \) responds less drastically.

Understanding \( k \) is crucial for interpreting how sensitive or responsive the dependent variable is to changes in the independent variables. Thus, the constant of variation acts as a bridge that connects the relational dynamics among the variables.

Dependent Variable

The dependent variable is a central concept in any variation model. In our equation, \( L = k m n \), \( L \) is identified as the dependent variable. This means the value of \( L \) relies on the values of the independent variables, \( m \) and \( n \), as well as the constant of variation, \( k \). When we say \( L \) "varies jointly," we mean it changes in response to the simultaneous effects of both \( m \) and \( n \).

• If either \( m \) or \( n \) increases, \( L \) will typically increase, assuming the other variables are constant.
• If either \( m \) or \( n \) decreases, \( L \) will typically decrease, under the same condition.

The beauty of the dependent variable is that it allows us to predict outcomes based on changes in other parts of the equation. Recognizing \( L \) as the dependent variable empowers us to understand the equation's dynamic nature.

Independent Variables

Independent variables are vital for understanding how variations occur in mathematical models. In the joint variation equation \( L = k m n \), the terms \( m \) and \( n \) are the independent variables. Their values can be changed to see how they affect the dependent variable, \( L \). What sets independent variables apart is:

• They do not rely on any other part of the equation for their values.
• They freely change, impacting the value of \( L \).

For example, if you adjust \( m \,\) keeping \( n \) constant, you can observe the variation in \( L \). Similarly, altering \( n \) with a constant \( m \) will show its effect on \( L \). The role of independent variables is to provide a base upon which the effects on the dependent variable can be studied. By controlling these inputs, we gain a clear insight into the relationship defined by the equation, \( L = k m n \).