Problem 42 - Intermediate Algebra

Question

Express each verbal model in symbols. \(d\) varies jointly as \(r\) and \(t\)

Step-by-Step Solution

Answer

The formula is \(d = krt\).

Step 1: Understand the Meaning of Joint Variation

Joint variation implies that one variable depends on two or more other variables simultaneously. In this case, it means that the variable \(d\) changes according to the product of \(r\) and \(t\).

Step 2: Write the General Formula for Joint Variation

The expression for joint variation includes a constant of proportionality \(k\). This gives the formula: \(d = krt\), where \(k\) is a constant.

Key Concepts

Constant of Proportionality

In mathematics, understanding the concept of the constant of proportionality is crucial, especially when dealing with variations. This constant, often represented by the symbol \( k \), is the factor that bridges the relationship between variables in a proportional relationship. For joint variation, such as \( d = krt \), the constant of proportionality \( k \) dictates how much one variable will change in response to changes in the other two variables simultaneously.

• The value of \( k \) remains constant for a given set of conditions.
• Without knowing \( k \), the exact relationship or magnitude of change between \( d \), \( r \), and \( t \) cannot be determined.
• Finding \( k \) often requires additional context or data, such as specific values for the involved variables.

By identifying \( k \), we can predict how changing \( r \) or \( t \) impacts \( d \). This makes the constant of proportionality an essential component in translating real-world scenarios into mathematical models.

Verbal Model

A verbal model in mathematics is a way to describe relationships using words rather than equations or symbols. It's about explaining the connection between variables in plain language. When we say "\( d \) varies jointly as \( r \) and \( t \)", it translates to a scenario where \( d \) is influenced by two factors at once.

• This model serves as a foundation for creating an algebraic expression that represents the relationship.
• Using words helps in visualizing and understanding the problem context before introducing mathematical notation.
• Verbal models are particularly useful in teaching contexts, where initial comprehension is built through language before abstraction is approached.

Such models provide clarity in understanding complex relationships and help in gradually transitioning to mathematical formulations, making them an integral part of problem solving.

Algebraic Expression

An algebraic expression utilizes symbols and operations to represent relationships between variables. Converting a verbal model like "\( d \) varies jointly as \( r \) and \( t \)" into the algebraic expression \( d = krt \) involves defining how the variables interact using mathematical terms. Here, it signifies a joint variation, a situation where one quantity depends on the multiplication of other quantities.

• The symbol \( d \) stands for the dependent variable or the result.
• \( r \) and \( t \) are independent variables that influence \( d \).
• \( k \) acts as the constant of proportionality, giving the expression its proper scale.

Algebraic expressions allow for precise calculations and predictions. They form the backbone of mathematical analysis, translating descriptive language into a format that can be manipulated, solved, and applied to real-world situations.