Problem 40 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) varies directly as the square root of \(x\) and inversely as \(z\) and the square of \(w,\) where \(y=27\) when \(x=9, w=1 / 2,\) and \(z=4\).

Step-by-Step Solution

Answer

The model is \( y = 9 \cdot \frac{\sqrt{x}}{z \cdot w^2} \).

Step 1: Understanding the Variation

The problem states that \(y\) varies directly as the square root of \(x\) and inversely as \(z\) and the square of \(w\). This means:\[ y = k \cdot \frac{\sqrt{x}}{z \cdot w^2} \]where \(k\) is a constant of proportionality that we need to find.

Step 2: Substitute Known Values

We are given \(y = 27\), \(x = 9\), \(w = \frac{1}{2}\), and \(z = 4\). Substitute these values into the equation to find \(k\):\[ 27 = k \cdot \frac{\sqrt{9}}{4 \cdot \left(\frac{1}{2}\right)^2} \]

Step 3: Simplify the Equation

Calculate \(\sqrt{9}\) and \(\left(\frac{1}{2}\right)^2\): \[ \sqrt{9} = 3 \]\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]Substitute these back into the equation:\[ 27 = k \cdot \frac{3}{4 \cdot \frac{1}{4}} \]

Step 4: Solve for k

Simplify the fraction \(\frac{3}{4 \cdot \frac{1}{4}}\):\[ \frac{3}{4 \cdot \frac{1}{4}} = \frac{3}{1} = 3 \]So the equation becomes:\[ 27 = k \cdot 3 \]Solving for \(k\), we get:\[ k = \frac{27}{3} = 9 \]

Step 5: Write the Mathematical Model

With \(k = 9\), substitute back into the formula for \(y\) to get the mathematical model:\[ y = 9 \cdot \frac{\sqrt{x}}{z \cdot w^2} \]

Key Concepts

Direct Variation

In the context of the problem, direct variation is when one variable changes in relation to another variable in such a way that their ratio remains constant. Here, the variable \(y\) varies directly with the square root of \(x\). The term *directly* implies that as \(x\) increases, \(y\) also increases, provided that other factors remain constant. This relationship can be expressed as:

• \(y = k \cdot \sqrt{x}\)

where \(k\) is the proportionality constant. This equation highlights that the change in \(y\) is consistent with the change in \(\sqrt{x}\). This kind of mathematical relationship is crucial as it allows predictions and calibrations, especially when dealing with proportional data.
It simplifies complex real-world scenarios into more digestible forms, making it easier to manipulate and understand variable dependencies.

Inverse Variation

Inverse variation is quite different from direct variation. It occurs when an increase in one variable leads to a proportional decrease in another variable. In simpler terms, the product of the two variables remains constant. In this exercise, \(y\) varies inversely with \(z\) and the square of \(w\). This relationship can be mathematically modeled as:

• \(y = \frac{k}{z \cdot w^2}\)

Here, \(k\) remains constant, but the equation indicates that as \(z\) or \(w\) increases, \(y\) will decrease, assuming all other variables stay constant. This concept is vital in scenarios such as electrical circuits (where current and resistance are inversely related) and can competently describe phenomena where one variable counteracts the impact of another.

Proportionality Constant

The proportionality constant \(k\) is the backbone of both direct and inverse relationships in mathematical modeling. This constant binds the relationship between variables, providing a stable ratio or factor. In our particular exercise, \(k\) was found through substituting known values into the derived equation to balance the proportional relationship:

• Substitute known values: \(27 = k \cdot \frac{3}{4 \cdot \frac{1}{4}}\)
• Calculate \(k\): \(k = 9\)

For students learning mathematical modeling, \(k\) is an essential part of understanding and creating equations that succinctly describe how variables interact with one another. Knowing how to find and use \(k\) reinforces understanding of proportionality and lays the foundation for more complicated analyses involving varying quantities.

Mathematical Modeling

Mathematical modeling is a powerful method of representing real-world phenomena and relationships through equations. This allows for prediction, analysis, and interpretation of data. In this exercise, the model is constructed to show how \(y\) behaves as a function of \(x\), \(z\), and \(w\):

• \(y = 9 \cdot \frac{\sqrt{x}}{z \cdot w^2}\)

This model not only reflects the interaction between the variables but also simplifies the complexity of those relationships. Through mathematical modeling, we transform abstract ideas into explicit formulas that provide clarity and insight, crucial for scientific, economic, and engineering studies. Learning how to create and interpret these models enables students to decipher the underlying patterns and structures governing various systems.