Problem 26 - Algebra 2

Question

Write the function that models each variation. Find \(z\) when \(x=4\) and \(y=9\) \(z\) varies directly with the square of \(x\) and inversely with \(y .\) When \(x=2\) and \(y=4, z=3 .\)

Step-by-Step Solution

Answer

The function that models each variation is \(z = 3x^2/y\), and for \(x=4\) and \(y=9\), \(z=16/3\).

Step 1: Determine Constant of Variation

To find the constant of variation, \(k\), use the information that when \(x=2\) and \(y=4\), \(z=3\). Then, substitute these values into the direct and inverse variation equation \(z = kx^2/y\), resulting in \(3 = k(2)^2/4\). Solving this equation for \(k\) gives \(k=3\).

Step 2: Write Variation Function

Using the solution for \(k\) from step 1, the function can be written as \(z = 3x^2/y\). This function will enable the calculation of \(z\) for any given \(x\) and \(y\) values.

Step 3: Find Z Value

Finally, use the function found in step 2 to calculate \(z\) when \(x=4\) and \(y=9\). Substitute these values into the function to get: \(z=3(4)^2/9\). Carry out the arithmetic operations to solve for \(z\).

Key Concepts

Constant of Variation

In mathematics, the concept of the constant of variation is often used in problems involving direct and inverse variation. This constant, typically represented as \(k\), provides a link between variables in an equation. In our exercise, we determined \(k\) by using given values where \(x = 2\), \(y = 4\), and \(z = 3\). These values were substituted into the equation \(z = \frac{kx^2}{y}\). After rearranging, we solved the equation:

• Substitute: \(3 = \frac{k \cdot (2)^2}{4}\)
• Solve for \(k\): \(12 = 4k \Rightarrow k = 3\)

This means that the constant of variation \(k\) equals 3, allowing the variation relationship to stay consistent across different values of \(x\) and \(y\). Understanding this constant is crucial, as it can transform abstract relationships into usable mathematical models.

Mathematical Modeling

Mathematical modeling is the process of developing an equation or function that represents a real-world scenario. In many variation problems, this involves combining direct and inverse variation into a single function. In the given exercise, the goal was to create a model describing how \(z\) varies with both the square of \(x\) and \(y\). The model is achieved with the formula:

• \(z = \frac{kx^2}{y}\)

Where for our case, \(k\) is found to be 3.This model acts as a predictive tool enabling us to calculate \(z\) for any suitable values of \(x\) and \(y\). Mathematical modeling is powerful because it turns theoretical concepts into practical solutions, providing essential insights into how different factors interact.

Function Writing

The process of function writing involves translating a verbal description into a mathematical equation or function. This step is crucial for solving variation problems, as it determines the mathematical relationship among variables.For this exercise, once we knew \(k\), we wrote the function:

• \(z = \frac{3x^2}{y}\)

This function compactly incorporates the direct variation with \(x^2\) and the inverse variation with \(y\). Finally, using this newly written function, we calculated \(z\) for the specific case where \(x = 4\) and \(y = 9\). Substituting these values gives:

• \(z = \frac{3 \times (4)^2}{9} = \frac{48}{9} \approx 5.33\)

Writing functions is essential for expressing dynamic mathematical relationships succinctly and enabling precise evaluations and predictions.