Problem 21 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) varies inversely as \(x,\) and \(y=2\) when \(x=1 / 8\).

Step-by-Step Solution

Answer

The mathematical model is \( y = \frac{1/4}{x} \).

Step 1: Understanding Inverse Variation

When a variable y varies inversely as x, it means that as x increases, y decreases proportionally, and vice versa. Mathematically, this relationship is expressed as \( y = \frac{k}{x} \), where k is a constant.

Step 2: Identify Given Values

From the problem, we know that \( y = 2 \) when \( x = \frac{1}{8} \). These values will help us find the constant k.

Step 3: Find the Constant of Variation

Substitute the given values into the inverse variation equation to find k. This gives: \( 2 = \frac{k}{\frac{1}{8}} \). To solve for k, multiply both sides by \( \frac{1}{8} \), resulting in \( k = 2 \times \frac{1}{8} = \frac{1}{4} \).

Step 4: Construct the Mathematical Model

Now that we have determined the constant of variation, the mathematical model can be written as \( y = \frac{1/4}{x} \). This represents the inverse variation relationship given in the problem.

Key Concepts

Mathematical Modeling

Mathematical modeling is a way to represent real-world situations using mathematical concepts and language. This helps us understand and solve problems by examining relationships, patterns, and trends. In this exercise, the model describes how the variables \( y \) and \( x \) relate through inverse variation.
The key is to use known values to properly set up the relationship, ensuring accurate predictions or explanations of behavior. Such models are everywhere, from physics to economics, making them an essential tool for analysis and decision-making.
Always start by identifying which kind of mathematical relationship (like inverse variation) is present in the real-life scenario you're modeling. This helps form the basis of a valid mathematical representation.

Constant of Variation

The constant of variation is a crucial concept when dealing with relationships between variables. In an inverse variation, this constant \( k \) defines how the variables are interrelated by determining the proportionality. In simple terms, it captures how a change in one variable affects the other.
In the given problem, you use known values \( y = 2 \) and \( x = \frac{1}{8} \) to calculate the constant \( k \) as follows:

• Start with the formula for inverse variation: \( y = \frac{k}{x} \).
• Substitute the known values: \( 2 = \frac{k}{\frac{1}{8}} \).
• Solve for \( k \) by rearranging the equation, resulting in \( k = \frac{1}{4} \).

Once you have the constant, it helps in constructing the mathematical model and predicting other outcomes in similar scenarios. The constant of variation remains persistent in any situation where the inverse relationship holds under the same conditions.

Direct and Inverse Variation

Understanding the difference between direct and inverse variation is key to modeling relationships. Direct variation implies a direct proportionality, where if one variable increases, the other does too. This is expressed as \( y = kx \).
Inverse variation, like in our exercise, suggests that one variable increases while the other decreases, as seen in the expression \( y = \frac{k}{x} \).
Some essential points to note:

• With direct variation, the ratio \( \frac{y}{x} \) stays constant.
• In inverse variation, the product \( y \times x \) stays constant.

These differences are pivotal because they change how we interpret data and predict outcomes. By mastering these concepts, students can more effectively translate real-world scenarios into mathematical models, allowing for greater analysis and insight into the problems they tackle.