Problem 6 - Algebra 2

Question

Suppose that \(x\) and \(y\) vary inversely. Write a function that models each inverse variation. $$ x=2.5 \text { when } y=100 $$

Step-by-Step Solution

Answer

The function modeling the inverse variation is \(x=\frac{250}{y}\).

Step 1: Understanding inverse variation

When two quantities vary inversely, their product is constant. Mathematically, this is represented as \(xy=k\), where \(k\) is a constant.

Step 2: Finding the constant of variation

To find the constant \(k\), substitute the given values of \(x\) and \(y\) into the equation. Hence, if \(x=2.5\) when \(y=100\), then \(k=xy=2.5*100\). Calculate this to find \(k\).

Step 3: Writing the function

Substitute this value of \(k\) back into the general equation \(xy=k\) to find the required function. Hence, the model inverse variation would be \(x=\frac{k}{y}\).

Key Concepts

Constant of Variation

In inverse variation, two variables are connected by a unique constant known as the "constant of variation," often represented by the symbol \( k \). This concept is based on the idea that the product of the two variables remains constant, no matter the individual values of these variables. For example, if \( x \) and \( y \) vary inversely, then their product \( xy \) is always equal to \( k \). To illustrate this, consider a scenario where \( x=2.5 \) and \( y=100 \). When these values are multiplied, the product \( 2.5 \times 100 \) will give us the constant \( k \). This means that no matter how \( x \) and \( y \) change, as long as they vary inversely, their product will always equal \( 250 \), maintaining the relationship established by the constant.

Understanding the constant of variation is crucial because it helps in forming the equation that represents the inverse relationship between the variables. With the constant known, you can easily express one variable in terms of the other.

Product of Variables

In the context of inverse variation, the key idea is that the product of the two variables involved will always equal the constant of variation. This means that as one variable increases, the other must proportionally decrease to keep their product unchanged. The relationship is modeled by the equation \( xy = k \), where \( x \) and \( y \) are the variables, and \( k \) is their constant product. By understanding this, you can predict changes in one variable when the other is altered. For instance, if \( y \) becomes larger, \( x \) must become smaller to ensure the product \( xy \) still equals \( k \). This reflects a fundamental property of inversely varying quantities.

In practical terms, if you know any two out of these three components — the values of \( x \), \( y \), or \( k \) — you can determine the third without additional information. This fact highlights the interconnected nature of the variables in inverse variation.

Algebraic Functions

Algebraic functions are expressions crafted using basic operations such as addition, subtraction, multiplication, and division, combined with variables and constants. In the context of inverse variation, algebraic functions help to describe and model the relationship between two inversely varying variables. For our example, the inverse variation function can be written as \( x = \frac{k}{y} \). This expression is a classic example of how algebraic functions utilize division to convey a variable relationship where as \( y \) increases, \( x \) decreases, confirming the inverse nature of the variation. These functions are pivotal because they allow clear and concise expression of complex relationships, providing a straightforward method to predict and calculate variable changes. In studying inverse variations, understanding these algebraic formulations empowers you to easily model real-world scenarios where such variations naturally occur, like speed and time, which maintain a consistent relationship through their interaction.