Problem 22 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) varies inversely as \(x,\) and \(y=3 / 2\) when \(x=1 / 9\).

Step-by-Step Solution

Answer

The model is \( y = \frac{1}{6x} \).

Step 1: Understanding Inverse Variation

In an inverse variation, two variables are related such that as one increases, the other decreases. Mathematically, this is represented by the equation \( y = \frac{k}{x} \), where \( k \) is the constant of variation.

Step 2: Identify Known Values

We are given that \( y = \frac{3}{2} \) when \( x = \frac{1}{9} \). Use these values to find the constant of variation \( k \).

Step 3: Calculate the Constant of Variation

Substitute the given values of \( y \) and \ and \( x \) into the inverse variation formula: \( \frac{3}{2} = \frac{k}{1/9} \). To find \( k \), multiply both sides by \( \frac{1}{9} \): \( k = \frac{3}{2} \times \frac{1}{9} = \frac{3}{18} = \frac{1}{6} \).

Step 4: Construct the Mathematical Model

With the constant \( k = \frac{1}{6} \), the model can be constructed as \( y = \frac{1/6}{x} \) or in simplified form \( y = \frac{1}{6x} \). This represents the relationship between \( y \) and \( x \) as an inverse variation.

Key Concepts

Constant of Variation

In an inverse variation, the concept of the constant of variation is central. It acts like the skeleton of the relationship between two variables. When we say that a variable varies inversely as another, it means that as one variable goes up, the other comes down. This relationship can be written using the formula: \( y = \frac{k}{x} \) Here, \( k \) is the constant of variation. It doesn't change as \( x \) and \( y \) change, making it a crucial value that holds the relationship intact. To find this constant, we use specific values given in the problem. For example, if \( y = \frac{3}{2} \) when \( x = \frac{1}{9} \), we can replace these values into the equation and find \( k \) by solving it algebraically. Once you find \( k \), it remains the same for all pairs of values of \( x \) and \( y \) in that particular variation relationship.

Mathematical Model

A mathematical model is like a bridge that connects theoretical math to real-world problems. It serves as a simplified version of reality that helps us understand complex systems. In the case of inverse variation, creating a mathematical model is about capturing the essence of the relationship between two variables in a neat formula. Given the inverse relationship: \( y = \frac{k}{x} \) Once we calculate the constant of variation \( k \), we can construct a complete model. For instance, if we find that \( k = \frac{1}{6} \), our model becomes \( y = \frac{1}{6x} \). This equation isn't just numbers and letters — it's a dynamic tool. It allows us to predict \( y \) for any value of \( x \), and vice versa, enabling practical applications like engineering calculations or economic forecasts.

Variable Relationship

Understanding the relationship between variables is key to solving many mathematical problems. In the context of inverse variation, the concept of variable relationship tells us how the values of two variables affect each other. Inverse variation means that when one variable increases, the other decreases. Consider the example where \( y = \frac{1}{6x} \). Here, \( x \) and \( y \) have a reciprocal relationship. If \( x \) gets larger, \( y \) will get smaller, and this happens proportionally based on our constant of variation, \( \frac{1}{6} \). Recognizing these patterns helps students or professionals anticipate and adjust outcomes based on variable changes effectively. It’s like understanding that in a seesaw, as one side goes up, the other must come down, so you can balance the whole system.