Problem 49 - Intermediate Algebra

Question

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ U=k r s^{2} t $$

Step-by-Step Solution

Answer

\( U \) varies jointly with \( r \), \( t \), and the square of \( s \).

Step 1: Identify the Variables and their Relationships

The given equation is \( U = k r s^2 t \). Here, \( U \) is a variable dependent on \( r \), \( s \), and \( t \). \( k \) is the constant of variation. The equation suggests that \( U \) varies with the product of \( r \), the square of \( s \), and \( t \). Your task is to express these relationships in words.

Step 2: Describe Each Part of the Equation

In the equation \( U = k r s^2 t \), notice the terms individually: \( r \) varies directly with \( U \), \( s \) influences \( U \) through its square, meaning \( U \) varies with the square of \( s \), and \( t \) also varies directly with \( U \). Combining these relationships, you can formulate how \( U \) varies.

Step 3: Write the Variation Model in Words

To express the variation model in words: \( U \) varies jointly with \( r \), \( t \), and the square of \( s \). This implies that increasing any of these variables while keeping the others constant will result in an increase in \( U \), and vice-versa.

Key Concepts

Constant of Variation

In mathematics, when dealing with joint variation, the constant of variation, denoted as \( k \), plays a crucial role. It is essentially what ties all the varying elements together in a variation equation. In our exercise, \( k \) is a fixed number that scales the relationship between the variables \( r \), \( s \), and \( t \) and the dependent variable \( U \).

The constant of variation can be thought of as a scaling factor. It determines the rate at which the dependent variable changes as the other variables change. For example, if \( k \) is larger, the rate at which \( U \) increases or decreases will be more substantial for the same change in \( r \), \( s \), or \( t \). On the flip side, if \( k \) is smaller, the impact on \( U \) due to changes in the other variables will be less dramatic.

• \( k \) provides consistency across different scenarios of the equation.
• This constant allows us to predict and adjust \( U \) based on changes in \( r \), \( s \), and \( t \).
• By knowing \( k \), calculations become manageable and outcomes predictable.

Dependent Variable

A dependent variable is a core concept in mathematical relationships where one variable depends on others. In our case, \( U \) is the dependent variable. This means its value is determined by the values of the independent variables \( r \), \( s \), and \( t \).

The equation \( U = k r s^2 t \) illustrates that \( U \) relies on these other variables to define its value. As such, any change in \( r \), \( s \), or \( t \) will directly impact \( U \). Understanding which variable is dependent helps in solving problems and making predictions based on given data.

• The value of \( U \) changes according to variations in \( r \), \( s \), and \( t \).
• It is important to know which variable in an equation is dependent to comprehend the dynamics of a model fully.
• Analyzing the dependent variable provides insights into how different factors influence outcomes.

Direct Variation

Direct variation is a relationship between two variables where one is a constant multiple of the other. In simpler terms, when one variable increases, the other increases in a direct proportion, and when one decreases, the other decreases likewise.

In the joint variation equation \( U = k r s^2 t \), direct variation can be seen in the relationship of \( U \) with \( r \) and \( t \). This implies that \( U \) increases in direct proportion to \( r \) and \( t \) when the other variables remain constant. The square of \( s \) adds another layer but maintains a kind of squared direct variation with \( U \).

• Direct variation shows a straightforward, linear relationship between two variables, simplified by the constant \( k \).
• This relationship allows for predictable results: if one variable doubles, so does the other, assuming the constant remains unchanged.
• It's essential for understanding how changes in one factor directly impact another.