Problem 27 - Algebra 2

Question

Write the function that models each variation. Find \(z\) when \(x=4\) and \(y=9\) \(z\) varies inversely with the product of \(x\) and \(y .\) When \(x=2\) and \(y=4, z=0.5\)

Step-by-Step Solution

Answer

The function that models this variation is \(z=4/(xy)\). When \(x=4\) and \(y=9\), \(z=1/9\).

Step 1: Understand inverse variation and formulate the function

An inverse variation is a relationship that can be written in the form \(z=k/xy\), where \(k\) is the constant of variation. From the given condition \(z=0.5\) when \(x=2\) and \(y=4\), we can substitute these values into the equation and solve for \(k\).

Step 2: Calculate the constant of variation

Substituting \(z=0.5\), \(x=2\), and \(y=4\) into the equation \(z=k/xy\), we get \(0.5 = k / (2*4)\). Solving this equation for \(k\), we find that \(k=4\). So, the function of the variation \(z\) with respect to \(x\) and \(y\) is \(z=4/(xy)\).

Step 3: Find the value of \(z\) for \(x=4\) and \(y=9\)

Now, with \(x=4\) and \(y=9\), the equation becomes \(z = 4/(4*9)\). Simplifying, we find that \(z = 1/9\).

Key Concepts

Algebraic Functions

Algebraic functions are expressions that involve variables and constants combined using algebraic operations such as addition, subtraction, multiplication, division, and exponentiation. In the context of inverse variation, the function is usually represented as a quotient of two algebraic expressions.
Inverse variation describes a relationship where one variable increases as the other decreases. This is expressed mathematically as \(z = \frac{k}{xy}\), where \(z\) is the variable that varies inversely with the product of \(x\) and \(y\), and \(k\) is a constant that does not change for a specific relation.

• The function models how one variable affects another inversely.
• As the product \(xy\) increases, \(z\) decreases, and vice versa.

Understanding the algebraic form helps in solving problems related to inverse variation as it provides a structured approach to find the constant and the varying values of the variables involved.

Constant of Variation

The constant of variation \(k\) in an inverse variation equation \(z = \frac{k}{xy}\) determines how strongly \(z\) responds to changes in \(x\) and \(y\). This constant is crucial as it remains the same regardless of the specific values of \(x\) and \(y\).
In our problem, by substituting the known values \(z = 0.5\), \(x = 2\), and \(y = 4\) into the function, we can solve for \(k\) as follows:

• Equation: \(0.5 = \frac{k}{2 \cdot 4}\)
• Solving gives \(k = 4\)

This value of \(k\) suggests the strength and direction of the variation between \(z\), \(x\), and \(y\). Once \(k\) is determined, it allows for predictions and calculations of \(z\) for any given \(x\) and \(y\) provided they are within the constraints of the scenario.

Problem Solving

Problem solving in algebra often involves breaking down a given scenario into manageable steps. For inverse variation problems, the initial task is often to establish the algebraic relationship as defined by the problem statement. This involves identifying the correct form of the equation and the variable relationship.
Here are some fundamental steps in solving inverse variation problems:

• Determine the form of the equation (inverse variation: \(z = \frac{k}{xy}\)).
• Plug in given values to solve for the constant \(k\).
• Use the calculated constant to find the desired variable under new conditions.

In our exercise, after calculating \(k\), we apply it to find \(z\) when \(x = 4\) and \(y = 9\):

• Equation: \(z = \frac{4}{4 \cdot 9}\)
• Solving gives \(z = \frac{1}{9}\)

This systemic approach simplifies complex variations into direct solutions, making them easier to handle.