Problem 18 - Algebra A Combined Function

Question

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=\frac{1}{10}\) when \(x=40\)

Step-by-Step Solution

Answer

The constant is 4, and the equation is \(y = \frac{4}{x}\).

Step 1: Understanding Inverse Variation

Inverse variation implies that when one variable increases, the other decreases. Mathematically, we say that \(y\) varies inversely with \(x\) if \(y = \frac{k}{x}\), where \(k\) is the constant of variation.

Step 2: Identifying Given Values

From the problem statement, we have \(y = \frac{1}{10}\) and \(x = 40\). We will use these values to find the constant \(k\).

Step 3: Finding the Constant of Variation

Substitute \(y\) and \(x\) into the inverse variation formula: \(\frac{1}{10} = \frac{k}{40}\). To find \(k\), multiply both sides by 40: \(k = 40 \, \cdot \, \frac{1}{10}\).

Step 4: Calculating Constant \(k\)

Perform the multiplication: \(k = \frac{40}{10} = 4\). So, the constant of variation is \(k = 4\).

Step 5: Writing the Inverse Variation Equation

Now that we have the constant \(k = 4\), substitute it back into the inverse variation equation: \(y = \frac{4}{x}\). This is the equation that describes the inverse variation relationship between \(y\) and \(x\).

Key Concepts

Understanding the Constant of Variation

In the world of inverse variation, the constant of variation plays a pivotal role. It's the fixed number that defines the relationship between two variables that are inversely related. When we say that a variable "varies inversely," it means that as one variable increases, the other decreases, and their product is constant.

To find this constant, we use the formula for inverse variation, which is given by:

• \( y = \frac{k}{x} \)

Here, \( k \) is the constant of variation. In our provided exercise, with \( y = \frac{1}{10} \) and \( x = 40 \), we identify \( k \) using the equation \( \frac{1}{10} = \frac{k}{40} \). Solving for \( k \) involves basic algebra: multiplying both sides by 40 gives \( k = 4 \).

That means, no matter how \( x \) and \( y \) change, as long as \( x \cdot y = 4 \), they're maintaining that inverse relationship defined by our constant.

Inverse Variation Equation Made Easy

The inverse variation equation is a simple yet powerful tool. It allows us to model situations where one quantity decreases as another increases, a common scenario in the real world.

The general form of an inverse variation equation is:

• \( y = \frac{k}{x} \)

where \( k \) represents the constant of variation that you've just found. In our scenario, the equation becomes \( y = \frac{4}{x} \).

This specific equation helps us quickly determine \( y \) for any value of \( x \). Similarly, if \( y \) is known, \( x \) can be calculated with ease. Such relationships are crucial in fields like physics to understand how one variable affects another, particularly when considering things like pressure and volume in gases which also exhibit inverse variation characteristics.

Exploring Mathematical Relationships Through Inverse Variation

Mathematical relationships describe how quantities are connected to one another. The inverse variation is one such relationship, indicating a unique connection where one variable decreases as the other increases.

There are various tell-tale signs of inverse variation:

• The graph of an inverse variation is typically a hyperbola.
• As one variable doubles, the other halves, keeping their product constant.

These characteristics can be handy when identifying if a real-world situation fits an inverse variation model.

For instance, in economics, cost can vary inversely with supply. As supply goes up, the cost tends to go down, assuming demand is constant. Such patterns recur across numerous fields, and understanding these mathematical relationships aids in dissecting and analyzing complex systems.