Problem 2 - Precalculus: Mathematics for Calculus

Question

If the quantities \(x\) and \(y\) are related by the equation \(y=\frac{3}{x}\) then we say that \(y\) is _______________ _______________ to \(x\) and the constant of _______________ is 3

Step-by-Step Solution

Answer

y is inversely proportional to x and the constant of proportionality is 3.

Step 1: Understand the Relationship

The problem describes a situation where the equation \( y = \frac{3}{x} \) is given. This is an indication of an inverse relationship between \( y \) and \( x \). In such a relationship, as one variable increases, the other decreases.

Step 2: Identify Proportionality Type

Since the equation is of the form \( y = \frac{k}{x} \), where \( k \,=\, 3 \), it implies that \( y \) is inversely proportional to \( x \). This type of relationship is expressed by saying "\( y \) is inversely proportional to \( x \)."

Step 3: Determine the Constant of Proportionality

In the equation \( y = \frac{k}{x} \), \( k \) is the constant of proportionality. Here, the constant \( k \) is given explicitly as 3, meaning it is the constant that relates \( y \) and \( x \) in this inverse relationship.

Key Concepts

Inverse Relationship

In an inverse relationship, two variables move in opposite directions. If one variable increases, the other decreases. This can be visualized easily through a seesaw analogy. Imagine a seesaw where increasing one side causes the other side to lower, and vice versa. Mathematically, an inverse relationship is represented by the equation \( y = \frac{k}{x} \), where \( k \) is a constant.

In the equation given, \( y = \frac{3}{x} \), the variables \( y \) and \( x \) are inversely related. As \( x \) grows larger, \( y \) becomes smaller. Understanding this concept helps in predicting how changes in one variable affect the other in real-world situations, like how speed affects travel time for a fixed distance.

Constant of Proportionality

The constant of proportionality is a key value in equations that express relationships between variables.

In both direct and inverse relationships, this constant helps to determine how changes in one variable will affect another. Specifically, in an inverse relationship described by the formula \( y = \frac{k}{x} \), \( k \) acts as the constant of proportionality.

• It stays the same, regardless of the values \( x \) and \( y \) take.
• It tells us how the relationship between the variables is structured.

In the example \( y = \frac{3}{x} \), \( k = 3 \) is the constant of proportionality.

It signifies that for the variations in \( x \) and \( y \), the product \( x \cdot y \) will always equal 3.

Proportionality

Proportionality describes relationships between quantities where one variable is a constant multiple of the other. There are two main types of proportionality – direct and inverse.

• **Direct Proportionality:** If a relationship is of the form \( y = kx \), then \( y \) is directly proportional to \( x \). Both variables increase or decrease together.
• **Inverse Proportionality:** As seen in our problem, if the relationship is \( y = \frac{k}{x} \), \( y \) is inversely proportional to \( x \). Here, one variable increases as the other decreases.

Understanding which type of proportionality applies is crucial for solving mathematical and real-world problems accurately.

It helps us determine how one quantity affects another and, with the constant of proportionality, provides a clear picture of the interaction between variables.