Problem 100 - College Algebra with Modeling and Visualization

Question

Assume that the constant of proportionality is positive. Let \(y\) vary inversely as the second power of \(x\). If \(x\) doubles, what happens to \(y ?\)

Step-by-Step Solution

Answer

When \( x \) doubles, \( y \) becomes \( \frac{1}{4} \) of its original value.

Step 1: Understanding Inverse Variation

Since \( y \) varies inversely as the second power of \( x \), we can express the relationship as \( y = \frac{k}{x^2} \), where \( k \) is the constant of proportionality.

Step 2: Initial Expression for y

If \( x = x_1 \), then \( y = \frac{k}{x_1^2} \). This gives us the formula in terms of the original \( x \).

Step 3: Doubling x

If \( x \) doubles, it becomes \( 2x_1 \). Substituting in the inverse variation formula, \( y = \frac{k}{(2x_1)^2} \).

Step 4: Simplifying the Expression

Simplifying the expression, we have \( y = \frac{k}{4x_1^2} \).

Step 5: Comparing the Two Expressions for y

Initially, \( y = \frac{k}{x_1^2} \). After doubling \( x \), \( y = \frac{k}{4x_1^2} \), which is \( \frac{1}{4} \) of the original value.

Key Concepts

Constant of Proportionality

When we say that a variable varies inversely with another variable, we mean that as one variable increases, the other decreases. The constant of proportionality, often denoted as \( k \), helps us understand the strength or intensity of this relationship.
In mathematical terms, when \( y \) varies inversely as the square of \( x \), the relationship is described by the formula \( y = \frac{k}{x^2} \). Here, \( k \) remains constant, no matter the values of \( x \) or \( y \), as long as the relationship holds true.
This constant is crucial because it ensures that the product \( y \times x^2 \) always remains constant, showing how closely and inversely one varies with the other.

Second Power

The second power, often called squaring, refers to multiplying a number by itself. In the formula \( y = \frac{k}{x^2} \), the second power of \( x \) plays a pivotal role.
By squaring \( x \), we essentially amplify its effect on \( y \). For instance, if \( x \) were to double, the term \( x^2 \) increases by a factor of four. This dramatizes the inverse relationship and shows how sensitive \( y \) is to changes in \( x \).
Understanding the second power is essential when analyzing inverse variation, as it affects how rapidly \( y \) decreases as \( x \) increases.

Inverse Variation Formula

Inverse variation describes a specific mathematical relationship between two variables, where one variable increases while the other decreases. The inverse variation formula is given by \( y = \frac{k}{x^2} \), where \( y \) and \( x \) are variables that vary inversely and \( k \) is the constant of proportionality.
This formula is a guide for determining how changes in one variable affect the other. When applying this formula, if \( x \) doubles, for example, then because \( x^2 \) becomes \( (2x)^2 = 4x^2 \), the value of \( y \) becomes one-fourth its original value.
This relationship shows the inverse proportionality in action, emphasizing how drastically \( y \) can change due to modifications in \( x \), especially when considering powers like \( x^2 \). This knowledge is vital for students to understand the dynamics of inverse relationships.