Problem 54 - Precalculus with Limits

Question

In Exercises 49-58, find a mathematical model for the verbal statement. \(z\) is jointly proportional to the square of \(x\) and the cube of \(y\).

Step-by-Step Solution

Answer

The mathematical model for the given statement is \(z = k x^2 y^3\)

Step 1: Understand Joint Proportionality

According to the problem, 'z' is jointly proportional to the square of 'x' and the cube of 'y'. This means that as we increase (or decrease) the values of 'x' and 'y', the value of 'z' will change as well in the same proportion.

Step 2: Translate Verbal Statement to Mathematical Model

The mathematical model of the statement will be \(z = k x^2 y^3\), where 'k' is the constant of proportionality that we do not know at this moment.

Step 3: Finish the Model

Therefore, the mathematical model for the given problem is \(z = k x^2 y^3\). In the case where we would have the values of 'x', 'y', and 'z', we could substitute them into this equation to solve for the constant 'k'.

Key Concepts

Constant of Proportionality

When dealing with proportional relationships, the constant of proportionality is an important factor. Initially, it may sound complex, but it's simply a fixed number that relates two quantities that are proportional to each other.
In our given problem, we say that the variable \( z \) is jointly proportional to \( x^2 \) and \( y^3 \). Here, the constant of proportionality is represented by \( k \).

• This constant remains unchanged unless specified otherwise, providing stability in calculations.
• In any relation like \( z = k x^2 y^3 \), \( k \) helps maintain the proportional equivalence between the variables as \( x \) and \( y \) change.

By substituting known values of \( x \), \( y \), and \( z \) into \( z = k x^2 y^3 \), we can solve for \( k \), thus defining the specific relationship between these variables in a given context.

Mathematical Modeling

Mathematical modeling is a method used to create a representation of real-world phenomena using mathematical language. It provides a structured approach to understanding complex statements or problems by translating them into equations.
In our exercise, a verbal statement "\( z \) is jointly proportional to the square of \( x \) and the cube of \( y \)" is translated into a mathematical model \( z = k x^2 y^3 \).

• The model captures the relationship between \( x \), \( y \), and \( z \) succinctly and accurately.
• It uses previously defined mathematical rules and concepts, like proportionality, exponents, and constants, to form a functional equation that can be manipulated to find solutions.

Mathematical models simplify complex problems, making it easier to predict and understand the behavior of variables under different conditions.

Exponents in Relation to Proportionality

Exponents play a critical role when modeling proportional relationships, especially when variables need to be multiplied by themselves several times. This process is known as exponentiation.
In the context of our expression \( z = k x^2 y^3 \), exponents indicate how many times the base number is used as a factor.

• Here, \( x^2 \) means \( x \times x \); it illustrates that the variable \( x \) influences \( z \) to a greater extent when squared compared to its base value.
• Similarly, \( y^3 \) suggests that \( y \) is used as a factor three times, indicating an even stronger effect on \( z \).

Understanding exponents in proportional relationships is crucial as it allows us to gauge the intensity of influence each variable has on the outcome.