Problem 39 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) varies directly as the square of \(x\) and inversely as \(z\) and the square of \(w,\) where \(y=14\) when \(x=4, w=2,\) and \(z=2\).

Step-by-Step Solution

Answer

The model is \( y = 7 \frac{x^2}{zw^2} \).

Step 1: Understand the relationship

The problem states that \( y \) varies directly as the square of \( x \) and inversely as \( z \) and the square of \( w \). This can be expressed as a mathematical equation: \( y = k \frac{x^2}{zw^2} \), where \( k \) is the constant of proportionality.

Step 2: Substitute known values

We know that \( y = 14 \), \( x = 4 \), \( z = 2 \), and \( w = 2 \). Substitute these values into the equation from Step 1: \( 14 = k \frac{4^2}{2 \times 2^2} \).

Step 3: Solve for the constant of proportionality

Simplify the fraction: \( 4^2 = 16 \) and \( 2^2 = 4 \). Now the equation is \( 14 = k \frac{16}{2 \times 4} = k \frac{16}{8} = k \cdot 2 \). Solve for \( k \) by dividing both sides by 2: \( k = 7 \).

Step 4: Construct the mathematical model

Now that we have \( k = 7 \), substitute it back into the generalized model equation: \( y = 7 \frac{x^2}{zw^2} \). This is the mathematical model describing the variation.

Key Concepts

Direct Variation

In mathematics, direct variation describes a simple relationship between two variables. If one variable increases or decreases, the other does so in step. For instance, if you double one variable, the other also doubles. This straightforward pattern is expressed with the equation:

• \( y = kx^n \)

where:

• \( y \) is the dependent variable,
• \( x \) is the independent variable,
• \( k \) is a non-zero constant, referred to as the constant of proportionality,
• \( n \) is the power to which \( x \) is raised.

In our specific example, the equation becomes \( y = kx^2 \), indicating that \( y \) increases with the square of \( x \). This means if \( x \) doubles, \( y \) grows fourfold (since 2 squared is 4). Direct variation is often used to model straightforward, predictable relationships.

Inverse Variation

Inverse variation works in the opposite way to direct variation. Here, increasing one variable results in the decrease of the other. Likewise, decreasing a variable causes an increase in the other.

• This relationship can be mathematically expressed as: \( y = \frac{k}{x^m} \).

This means that as one variable \( x \) increases, \( y \) decreases as it's divided by \( x^m \). For our example, \( y \) varies inversely as both \( z \) and \( w^2 \) (the square of \( w \)). This can be seen in the equation:

• \( y = k \frac{x^2}{zw^2} \).

To put it simply, as \( z \) or \( w \) rise, evaluating the equation shows that \( y \) shrinks. This inverse relationship is essential for modeling situations where two variables operate in contrast to each other.

Proportionality

Proportionality connects two variables in a uniform way. It ensures when one variable changes, the change in another variable is predictable and consistent. There are two main types of proportionality:

• **Direct proportionality**, where variables increase or decrease proportionally. In our case, it means \( y \) is directly proportional to \( x^2 \).
• **Inverse proportionality**, where an increase in one variable causes a proportional decrease in the other, observed with \( y \)'s relation to \( z \) and \( w^2 \).

Proportional relationships allow simplifying complex real-world problems, transforming them into easier-to-understand mathematical models. This understanding underpins creating useful models for everything from physics to economics.

Constant of Proportionality

The constant of proportionality, denoted as \( k \), is a crucial part of understanding variation. In any variation equation, the constant helps bind the relationship between the dependent and independent variables.

• For direct variation, the constant shows how much \( y \) changes as \( x \) changes.
• For inverse variation, it indicates how rapidly \( y \) adjusts when \( x \) shifts.

In the given example, the constant of proportionality was found to be \( k = 7 \), meaning for every change in \( x \) and \( z \) or \( w \), \( y \) changes consistently by a factor of 7.

• Effectively, \( k \) normalizes any unit discrepancies, ensuring the equation remains valid regardless of the number values plugged into it.

Finding \( k \) is a critical step when constructing and verifying mathematical models, as it bolsters the stability and reliability of the model's predictions.