Problem 36 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) varies jointly as \(x\) and \(z\) and inversely as the square of \(w,\) where \(y=5\) when \(x=1, z=3,\) and \(w=1 / 2\).

Step-by-Step Solution

Answer

The mathematical model is \(y = \frac{5}{12} \cdot \frac{xz}{w^2}\).

Step 1: Understanding the Problem

We are asked to construct a mathematical model where the variable \(y\) varies jointly as \(x\) and \(z\), and inversely as the square of \(w\). This means that \(y\) is proportional to \(x \cdot z\), and also inversely proportional to \(w^2\).

Step 2: Setting Up the Equation

To express the relationship, we use the form \(y = k \cdot \frac{xz}{w^2}\), where \(k\) is the constant of proportionality. This covers both the joint variation with \(x\) and \(z\), and the inverse variation with \(w^2\).

Step 3: Substitute Given Values

Substitute the given values into the equation to find \(k\). We substitute \(y = 5\), \(x = 1\), \(z = 3\), and \(w = \frac{1}{2}\) into \(5 = k \cdot \frac{1 \cdot 3}{(1/2)^2}\).

Step 4: Calculating the Constant

Solve for \(k\) by calculating the expression. We have \(5 = k \cdot \frac{3}{1/4} = k \cdot 12\). Thus, \(k = \frac{5}{12}\).

Step 5: Writing the Final Model

Substitute \(k\) back into the equation to get the final mathematical model: \(y = \frac{5}{12} \cdot \frac{xz}{w^2}\).

Key Concepts

Inverse Variation

Inverse variation is a concept where one quantity increases as another decreases proportionally. In the exercise, this is seen in how the variable \( y \) is inversely proportional to the square of \( w \). This inverse relationship can be expressed through multiplication of one value and division of another. The basic form for inverse variation is \( y = \frac{k}{x} \), where \( k \) is a constant.

• Here, \( y \) changes inversely with \( w^2 \). In simple terms, as \( w \) increases, \( y \) decreases, assuming \( x \) and \( z \) are constant.
• Inverse variation is crucial in modeling real-world scenarios where one variable negatively affects another.

Understanding inverse variation helps in identifying how changes in one part of a system can impact others, showing a balancing effect.

Proportionality Constant

The proportionality constant, denoted as \( k \), represents the fixed ratio that relates the varying quantities in a mathematical model. It allows you to equate two ratios in a proportional relationship. In this exercise, the constant \( k \) ties together the variations of \( y \) with \( x \, z \), and \( w^2 \).

• Determining \( k \) involves substituting known values into the variable equation \( y = k \cdot \frac{xz}{w^2} \).
• Solving for \( k \) provides a specific equation that accurately models the situation, concluding with \( k = \frac{5}{12} \).

Such constants are pivotal in constructing accurate models, reflecting the underlying consistent factors across different scenarios.

Mathematical Modeling

Mathematical modeling refers to creating equations and formulas to represent complex systems and situations. In this context, the model \( y = \frac{5}{12} \cdot \frac{xz}{w^2} \) expresses how different variables interact jointly and inversely. This is significant in various fields like science and economics for predicting and understanding behavior.

• It involves recognizing and translating a situation into a mathematical form, enabling analysis and conclusions.
• Models abstract real-world phenomena to find solutions instead of just observational insights.

Creating models requires comprehension of mathematical relationships, ensuring parameters accurately reflect the actual situation at hand.

Algebraic Equations

Algebraic equations are mathematical statements indicating the equality of two expressions. They form the core of mathematical modeling and problem-solving. In this exercise, the equation \( y = \frac{5}{12} \cdot \frac{xz}{w^2} \) connects variables in a structured format.

• Such equations are essential for solving problems by manipulating symbols and constants.
• They offer a precise language for expressing relationships and facilitating calculations.
• Algebraic equations also allow for easily updating models when new data is available, making them adaptive and robust tools in analysis.

Using these equations, you can evaluate various scenarios by adjusting values of variables, gaining insights, and planning accordingly.