Problem 89 - A Graphical Approach to Precalculus with Limits

Question

Assume that the constant of variation is positive. Let \(y\) be inversely proportional to \(x\). If \(x\) doubles, what happens to \(y ?\)

Step-by-Step Solution

Answer

If \(x\) doubles, \(y\) is halved.

Step 1: Understand the Concept of Inverse Proportion

In an inverse variation, the relationship between two variables \(y\) and \(x\) is such that the product of the two variables is constant. This can be expressed as \(y = \frac{k}{x}\) where \(k\) is the constant of variation.

Step 2: Initialize the Given Scenario

We start with an equation of inverse variation: \(y = \frac{k}{x}\). It's given that if \(x\) doubles, we need to determine how \(y\) behaves. Initially, assume \(x = x_0\) and \(y = y_0\) such that \(y_0 = \frac{k}{x_0}\).

Step 3: Modify the Variable \(x\)

According to the problem, \(x\) is doubled. This means the new value of \(x\) becomes \(2x_0\). Substitute \(2x_0\) into the inverse proportionality equation: \(y = \frac{k}{2x_0}\).

Step 4: Analyze the Change in \(y\)

Compare the initial value of \(y\) which is \(y_0 = \frac{k}{x_0}\) to the new value of \(y\), which is \(y = \frac{k}{2x_0}\). Simplifying the new expression for \(y\), \(\frac{k}{2x_0} = \frac{1}{2}\left(\frac{k}{x_0}\right)\).

Step 5: Conclusion

From Step 4, the new value of \(y\) is half the initial value \(y_0\). Thus, if \(x\) doubles, the value of \(y\) is halved. This is consistent with the properties of inverse variation.

Key Concepts

Proportions

In inverse variation, the concept of proportions is crucial for understanding the relationship between two variables. When two variables, say \(y\) and \(x\), are inversely proportional to each other, their relationship can be explained through a specific type of proportion.

• Here, the product of the two variables remains constant.
• This relationship can be represented by the equation \(y \cdot x = k\), where \(k\) is a fixed number known as the constant of variation.

In simpler terms, if one variable increases, the other decreases in such a way that their product stays the same. For example, if \(x\) doubles, to maintain the constant product, \(y\) must halve. This is because the proportion of change in \(x\) must exactly counterbalance the change in \(y\). Because of this interdependency, understanding proportions helps us predict changes in one variable when the other changes.

Constant of Variation

The constant of variation, denoted as \(k\), is the fundamental element that keeps the relationship between the variables steady in inverse variation.

• The constant \(k\) signifies this unchanging product in the relationship \(y = \frac{k}{x}\).
• It acts as a balancing factor for the variables as they adjust their values inversely.

In the problem, as you apply the inverse variation concept to practical scenarios, \(k\) remains unchanged regardless of how \(x\) or \(y\) fluctuate. For instance, when \(x\) becomes twice its original value, the expression for \(y\) transforms to \(y = \frac{k}{2x_0}\), showing that \(k\) stays constant while \(y\) changes according to the inverse relationship.Remember, recognizing the constant of variation helps us to see the consistent nature of the product of two variables and how this impacts their inverse relationship.

Variable Relationships

Understanding variable relationships in inverse variation involves recognizing how changing one variable impacts the other.

• In inverse variation, the relationship implies that moments of change in one is precisely met with an opposite change in the other.
• Mathematically expressed, you see this as \(y = \frac{k}{x}\), where any change in \(x\) affects \(y\) inversely.

For example, our exercise showed how doubling \(x\) resulted in halving \(y\). This predictable alteration helps us anticipate how variables will adjust to maintain the upright principle of a constant product \(k\). Understanding this helps to foresee outcomes when manipulating variable values and assists in solving complex mathematical problems efficiently. In summary, constantly analyzing the interaction between variables aids in developing a deeper comprehension not just for solving exercises, but in real-world situations where these principles are often observed.