Problem 76 - Elementary Algebra

Question

Construct a mathematical model given the following. $$ y \text { varies inversely as } x, \text { and } y=2 \text { when } x=5 $$

Step-by-Step Solution

Answer

The model is \( y = \frac{10}{x} \).

Step 1: Understand the Relationship

An inverse variation relationship means that as one variable increases, the other decreases. The mathematical representation of this is: \( y = \frac{k}{x} \), where \( k \) is a constant.

Step 2: Substitute Known Values

Given the specific values where \( y = 2 \) and \( x = 5 \), substitute these into the inverse variation equation to find \( k \): \( 2 = \frac{k}{5} \).

Step 3: Solve for the Constant \( k \)

To find \( k \), multiply both sides of the equation by 5: \( k = 2 \times 5 = 10 \).

Step 4: Write the Mathematical Model

Substitute \( k = 10 \) back into the inverse variation equation: \( y = \frac{10}{x} \). This is the mathematical model that describes the inverse relationship between \( y \) and \( x \).

Key Concepts

Mathematical Model

In mathematics, a model is an equation or formula that represents how different variables relate to each other. These models simplify real-world situations into mathematical terms. For instance, an equation can model how the speed of a vehicle depends on time or how supply and demand affect each other in economics. Here, we are dealing with a specific type of mathematical model called inverse variation. This involves two variables that have an inverse relationship, meaning they change in opposite ways. As one goes up, the other comes down. The model is represented by the equation \( y = \frac{k}{x} \). In this equation:

• \( y \) and \( x \) are variables that change in opposite directions.
• \( k \) is a constant that doesn't change.

Building such a model helps us predict how changes in one variable affect the other. When using a mathematical model like this, you plug in known values of \( y \) and \( x \) to solve for \( k \), the variation constant, allowing you to understand the relationship thoroughly.

Variation Constant

The variation constant, denoted as \( k \) in the context of inverse variation, is a crucial component of the mathematical model. It is the value that links the two variables together in the equation \( y = \frac{k}{x} \). In simpler terms, \( k \) makes the equation work to show how \( y \) and \( x \) are inversely related. To find \( k \), we substitute known values of \( y \) and \( x \) into our inverse variation formula.For example, in our problem, we know that \( y = 2 \) when \( x = 5 \). Plugging these numbers into \( 2 = \frac{k}{5} \) lets us solve for \( k \) as follows:

• Multiply both sides by 5 to isolate \( k \).
• \( k = 2 \times 5 = 10 \).

This means that the constant \( k \) is 10, which completes our mathematical model. By identifying \( k \), we can now use \( y = \frac{10}{x} \) to predict the value of one variable given the other at any point.

Inverse Relationship

An inverse relationship between two variables means that when one variable increases, the other decreases. This is different from a direct relationship, where both variables increase or decrease together. Imagine a seesaw: when one side goes up, the other must go down. This is similar to how variables interact in an inverse relationship.In the context of the mathematical model \( y = \frac{k}{x} \):

• If \( x \) increases, \( y \) decreases given a constant \( k \).
• Conversely, if \( x \) decreases, \( y \) increases.

Such a relationship is common in various scientific equations and practical scenarios, like the intensity of light inversely varying with distance from the source. Understanding inverse relationships are important as they help predict and explain behaviors in natural and social phenomena, illustrating how systems adjust as conditions change.