Problem 11 - Algebra and Trigonometry

Question

Write an equation that expresses the statement. \(z\) is proportional to the square root of \(y\)

Step-by-Step Solution

Answer

The equation is \(z = k \sqrt{y}\), where \(k\) is a constant.

Step 1: Understand the Concept of Proportionality

When we say that one variable is proportional to another, it means one can be expressed as a constant multiplied by the other. In this context, it implies that there exists a constant, let's call it "k," such that the relationship between them can be described with an equation.

Step 2: Identify the Variables

In the given problem, we have two variables. These are: \(z\), which is stated to be the variable proportional to something, and \(y\), which is involved in the square root function. We are interested in the relationship between \(z\) and \(\sqrt{y}\).

Step 3: Write the Proportionality Equation

Given that \(z\) is proportional to the square root of \(y\), we use the constant of proportionality, \(k\), to express this relationship in equation form. Hence, the equation is \(z = k \sqrt{y}\).

Step 4: Conclude the Equation Setup

The equation \(z = k \sqrt{y}\) succinctly captures the statement given. Here, \(k\) represents the constant of proportionality and \(\sqrt{y}\) is the mathematical operation applied to \(y\) in the relationship.

Key Concepts

Understanding Square Roots in Proportionality

A square root is a mathematical operation that finds a number, which when multiplied by itself, gives a specified number. In our exercise, we consider the expression \(\sqrt{y}\). This represents the square root of the variable \(y\). Here is an important note: every positive number actually has two square roots—a positive and a negative one. However, in most cases, we refer to only the positive square root. The square root is significant in proportionality problems because it allows us to express more complex relationships between variables. For example, if a variable \(z\) is proportional to \(\sqrt{y}\), it means that as \(y\) changes, \(z\) changes in a manner that maintains a constant ratio with the square root of \(y\). Understanding this relationship is key in correctly setting up the proportionality equation.

Exploring the Constant of Proportionality

The constant of proportionality, often denoted by \(k\), is a crucial concept in forming proportional relationships.

• It serves as the necessary factor that binds the two variables in a proportional relationship.
• In the equation \(z = k \sqrt{y}\), \(k\) represents the constant that directly links \(z\) to \(\sqrt{y}\).

This constant can take any real value depending on the specific conditions of a situation. The beauty of the constant of proportionality is how it simplifies complex relationships into straightforward formulas. Recognizing and calculating this constant can help solve a variety of quantitative problems, making it an essential part of mathematics.

Working with Variables in Equations

Variables are symbols that represent unknown values and are fundamental in forming equations. In our context, the variables \(z\) and \(y\) represent two different quantities. The variable \(z\) changes in response to \(y\), according to a defined relationship.

• The role of variables in equations like \(z = k \sqrt{y}\) is to encapsulate this relationship mathematically.
• Through variables, we can express general relationships and solve for unknowns.

When solving for one variable in terms of another, it is vital to grasp what each variable signifies and how they interact within the equation. By focusing on the relationship between \(z\) and \(y\), we gain insights into how one can be determined by knowing the other, within the scope of their proportionality.