Problem 51 - Precalculus with Limits

Question

In Exercises 49-58, find a mathematical model for the verbal statement. \(y\) varies inversely as the square of \(x\).

Step-by-Step Solution

Answer

The mathematical model for the statement 'y varies inversely as the square of x' is \(y = k / x^2\).

Step 1: Understand the formula for inverse variation

When \(y\) varies inversely as \(x\), it can be represented by the equation \(y = k / x\) where \(k\) is the constant of variation. However, in this case, \(y\) varies inversely as the square of \(x\), so the equation will be \(y = k / x^2\).

Step 2: Formulate the equation

Since no specific values are given for \(y\) and \(x\) to find \(k\), the mathematical model for the statement 'y varies inversely as the square of \(x\)' becomes \(y = k / x^2\).

Key Concepts

Variation Equations

Variation equations describe the relationship between two variables. There are different types of variation equations, including direct, inverse, joint, and combined variation.

For an inverse variation equation, one variable increases while the other decreases. In simpler terms, when one gets larger, the other gets smaller. For example, imagine a seesaw: if one side goes up, the other goes down. To express this mathematically, we use the equation format:

• General inverse variation: \( y = \frac{k}{x} \)
• Specific to our problem, where \(y\) varies inversely with the square of \(x\): \( y = \frac{k}{x^2} \)

Here, \(k\) is a constant, which doesn't change as \(x\) or \(y\) change.

Constant of Variation

The constant of variation, represented by \(k\), is a key component in variation equations. It acts like a bridge between the variables \(x\) and \(y\). While \(x\) and \(y\) change with respect to each other, \(k\) remains fixed. This fixed nature allows us to write dependable equations.

In the case of inverse variation, \(k\) is calculated by multiplying one variable by the inverse (or reciprocal) of the other variable. So when \(y = \frac{k}{x^2} \), you could rearrange to find \(k\) if values for \(y\) and \(x\) were provided:

• \(k = y \times x^2\)

Without specific values, \(k\) remains abstract, serving as a placeholder for relations.

Mathematical Modeling

Mathematical modeling involves crafting equations that represent real-world phenomena. It's a method for predicting behavior or outcomes of different systems. This process translates verbal descriptions into mathematical terms, allowing us to solve problems efficiently.In this exercise, the statement "\(y\) varies inversely as the square of \(x\)" is transformed into an equation. This equation, \(y = \frac{k}{x^2}\), represents a real-world scenario where understanding how \(y\) changes in relation to \(x\) is crucial.

By developing such models, we can:

• Predict how changes in \(x\) affect \(y\) without measuring \(y\) repeatedly.
• Use the model to compare different systems or scenarios logically.

Overall, mathematical modeling makes complex relationships easier to grasp and more applicable.