Problem 17 - Algebra A Combined Function

Question

If \(y\) varies inversely as \(x\), find the constant of variation and the inverse variation equation for each situation. \(y=\frac{1}{8}\) when \(x=16\)

Step-by-Step Solution

Answer

The constant of variation is 2, and the inverse variation equation is \( y = \frac{2}{x} \).

Step 1: Understand Inverse Variation

Inverse variation means that as one quantity increases, the other decreases such that the product of the two quantities remains constant. In other words, if \( y \) varies inversely as \( x \), then \( y = \frac{k}{x} \) for some constant \( k \).

Step 2: Calculate the Constant of Variation

Given that \( y = \frac{1}{8} \) when \( x = 16 \), we substitute these values into the inverse variation formula: \( \frac{1}{8} = \frac{k}{16} \). To find \( k \), we solve for \( k \) by multiplying both sides by 16: \( k = 16 \times \frac{1}{8} = 2 \).

Step 3: Write the Inverse Variation Equation

Now that we have found the constant of variation \( k = 2 \), the inverse variation equation based on this information is \( y = \frac{2}{x} \).

Key Concepts

Constant of Variation

In the context of inverse variation, the constant of variation is a special value that helps maintain the relationship between two inversely related variables. It is denoted by the symbol \( k \). When we say that two variables, for example \( y \) and \( x \), vary inversely, it means their product is consistent, or constant.

- For instance, if \( y = \frac{1}{8} \) when \( x = 16 \), the constant of variation \( k \) can be found by multiplying these two values. Using the formula \( k = y \times x \), we substitute to get \( k = \frac{1}{8} \times 16 \).

- To solve this: \( k = \frac{16}{8} = 2 \).

Thus, \( k \) is 2, which is the constant value that relates \( y \) and \( x \) within the given inverse variation relationship.

Inverse Variation Equation

The inverse variation equation is a simple yet powerful formula that captures the inverse relationship between two variables. If \( y \) varies inversely as \( x \), we express this as \( y = \frac{k}{x} \), where \( k \) is the constant of variation.

- In our given exercise, after calculating the constant of variation as \( k = 2 \), the inverse variation equation becomes \[ y = \frac{2}{x} \].

- This equation tells us that for any value of \( x \), we can quickly find \( y \) by dividing 2 by \( x \). Likewise, if we know \( y \) and need to find \( x \), we simply multiply \( y \) by \( x \) to find the constant \( k \), or rearrange the formula to solve for \( x \).

Relationship between Variables

Inverse variation describes a unique relationship between two variables where if one variable increases, the other must decrease so that their product remains consistent. This is in stark contrast to direct variation, where both variables move in the same direction.

- For example, if we consider the exercise where \( y = \frac{1}{8} \) for \( x = 16 \), and we know \( k = 2 \), then increasing \( x \) would decrease \( y \) to maintain the equation \( y = \frac{2}{x} \).

- This relationship is beneficial in various real-world situations, such as determining speed and time for a set distance, where increasing speed decreases travel time, keeping distance constant. Understanding this relationship aids in predicting how changes in one variable affect the other.