Problem 1 - Algebra 2

Question

Suppose that \(x\) and \(y\) vary inversely. Write a function that models each inverse variation. $$ x=1 \text { when } y=11 $$

Step-by-Step Solution

Answer

The function that models the inverse variation is \(y = 11/x\).

Step 1: Identify the given values

In the problem, we're given that \(x = 1\) when \(y = 11\). These are the known values that we will use to find the constant of inverse variation \(k\).

Step 2: Set up the inverse variation formula

The variables \(x\) and \(y\) vary inversely, therefore, their relationship can be written in the form of the equation \(xy = k\), where \(k\) is the constant of variation.

Step 3: Substitute the known values into the equation

Substitute \(x = 1\) and \(y = 11\) into the equation \(xy = k\). Solve for \(k\). Therefore, \(k = x \cdot y = 1 \cdot 11 = 11\).

Step 4: Write the Inverse Variation Function

Now having found \(k = 11\), substitute this value of \(k\) into the model formula \(y = k/x\) to get our inverse variation function, which hence becomes \(y = 11/x\). This function represents how the variables \(x\) and \(y\) vary inversely.

Key Concepts

Function Modeling

Function modeling helps us understand the relationship between different variables in mathematical terms. In this case, we're dealing with inverse variation, which occurs when two variables, like \(x\) and \(y\), change in such a way that as one increases, the other decreases. By modeling this relationship with a function, we can accurately predict how one variable changes in response to changes in the other.For inverse variation, the function is written as \(y = \frac{k}{x}\), where \(k\) is a constant value. This function tells us that the product of \(x\) and \(y\) always equals \(k\). By understanding how to model this relationship, we can explore various scenarios and solve problems involving inverse variation.

Constant of Variation

The constant of variation is the key to understanding inverse relationships. In our example, the constant of variation \(k\) is found by multiplying \(x\) and \(y\). We are given that \(x = 1\) when \(y = 11\), so we find \(k\) as follows:

• Substitute the values into the formula: \(xy = k\).
• Calculate: \(1 \times 11 = 11\).

This means \(k=11\). `k` is essential because it remains unchanged for any pair of values \(x\) and \(y\) in this inverse variation. No matter what \(x\) is, multiplying \(x\) by \(y\) will always give \(11\). Understanding this constant helps us grasp how inversely proportional relationships maintain consistency despite variable changes.

Inverse Relationships

Inverse relationships illustrate how two variables are connected in such a way that their product is constant. With inverse variation, as one variable increases, the other must decrease so that the product \(xy = k\) remains the same.Let's break it down:

• The relationship \(y = \frac{k}{x}\) means that multiplying \(x\) and \(y\) will always result in \(k\).
• If \(x\) increases, \(y\) must decrease to keep their product constant, and vice versa.

This type of relationship is common in real-life situations where resources are inversely related, like speed and time for a constant distance.By understanding inverse relationships, we can solve problems and make predictions about how changes in one quantity affect another.