Problem 50 - Algebra 2

Question

Suppose that \(x\) and \(y\) vary inversely. Write a function to model inverse variation. \(x=7.5\) when \(y=50\)

Step-by-Step Solution

Answer

The function modeling inverse variation in this case is \(y = 375/x\).

Step 1: Understand Inverse Variation

When \(x\) and \(y\) show inverse variation, it means when one value increases, the other decreases proportionally. This can be represented by a mathematical equation \(xy=k\), where \(k\) is a constant.

Step 2: Substitution

Now substitute \(x = 7.5\) and \(y = 50\) into the equation. So, \(7.5 \times 50 = k\). This comes out to \(k = 375\). Thus, the constant of variation in this particular case is \(375\).

Step 3: Write the Function

Now you can write the inverse variation function with \(k = 375\). The function becomes \(xy = 375\). This can be arranged to \(y = 375/x\), which represents the variation of \(y\) with respect to \(x\). This is the required function.

Key Concepts

Inverse Variation Formula

Inverse variation describes a situation where two variables, say \(x\) and \(y\), are related in such a way that when one variable increases, the other variable decreases in a proportional manner. This relationship is often represented by the equation \(xy = k\), where \(k\) is the constant of variation.
For example, if you were given the initial values of \(x = 7.5\) and \(y = 50\), you could find \(k\) by substituting these values into the equation, giving you \(7.5 \times 50 = k\). With this, \(k = 375\). The equation \(xy = 375\) becomes the model for this inverse variation, and can be rearranged to give the function \(y = \frac{375}{x}\).
So, when dealing with inverse variation, always remember this formula:

• \(xy = k\) to find the constant \(k\)
• \(y = \frac{k}{x}\) to express one variable in terms of the other

Constant of Variation

The constant of variation \(k\) plays a crucial role in the equation of inverse variation. It signifies the constant product of the variables \(x\) and \(y\) across their changes.
In our example, by substituting \(x = 7.5\) and \(y = 50\) into the formula, \(7.5 \times 50 = k\), we discover that \(k = 375\). This value of \(k\) remains unchanged even as the values of \(x\) and \(y\) vary.
Understanding the constant of variation is key to comprehending how the variables interact. When you know \(k\), you can easily predict the behavior of one variable given the other. The constant represents fixed relationships that provide a powerful way to model the dynamics between \(x\) and \(y\).
Keep in mind:

• The constant \(k\) does not change.
• It defines the mathematical model of the inverse variation.

Mathematical Modeling

Mathematical modeling in inverse variation helps us understand and predict behaviors of real-world phenomena where two variables exhibit inverse relationships. For instance, this is seen in instances such as speed and travel time, where increasing speed decreases travel time and vice versa.
Using the framework of inverse variation, we begin by setting up the model \(xy = k\). Identifying values like \(x = 7.5\) and \(y = 50\) lets us calculate the constant \(k = 375\), which forms the core of our model. The function then becomes \(y = \frac{375}{x}\), which vividly shows how changes in \(x\) will affect \(y\).
Through mathematical modeling, you can:

• Simulate various conditions and scenarios.
• Convey complex dynamics in a simple and understandable format.
• Facilitate problem-solving and decision-making processes in various fields.

With practice, mathematical modeling becomes an intuitive tool to represent and solve real-life situations.