Problem 18 - Precalculus Mathematics for Calculus

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. \(z\) varies inversely as \(t .\) If \(t=3,\) then \(z=5\).

Step-by-Step Solution

Answer

The equation is \(z = \frac{15}{t}\) with \(k = 15\).

Step 1: Identify the Relationship

The statement says that \(z\) varies inversely as \(t\), which means \(z\) is inversely proportional to \(t\). This is expressed mathematically as \(z \propto \frac{1}{t}\).

Step 2: Convert to Equation Form

To express the inverse variation as an equation, replace the proportionality with an equation: \(z = \frac{k}{t}\), where \(k\) is the constant of proportionality.

Step 3: Substitute Given Values

Use the provided values to find \(k\). Substitute \(z = 5\) and \(t = 3\) into the equation. This gives us the equation: \(5 = \frac{k}{3}\).

Step 4: Solve for the Constant

Solve the equation \(5 = \frac{k}{3}\) to find \(k\). Multiply both sides by 3 to isolate \(k\): \(k = 5 \times 3 = 15\).

Step 5: Write the Final Equation

Substitute \(k = 15\) back into the equation \(z = \frac{k}{t}\). Thus, the final equation is \(z = \frac{15}{t}\).

Key Concepts

Understanding the Constant of Proportionality

In the realm of mathematics, the constant of proportionality plays a crucial role in defining the relationship between two variables. When dealing with inverse variations, this constant, usually denoted as \(k\), helps articulate how one variable changes relative to another. In the exercise provided, since \(z\) varies inversely as \(t\), we set up the relation as \(z = \frac{k}{t}\). Here, \(k\) is our constant of proportionality.

The constant \(k\) remains constant (hence the name) for any pair of \(z\) and \(t\) that satisfy the original relationship. Once we substitute particular values for \(z\) and \(t\) into our equation, we can solve for \(k\). Understanding and calculating \(k\) accurately is vital, as it defines the entire behavior of the inverse relationship.

Formulating Mathematical Equations

Mathematics is a language of numbers, and equations are one of its core constructs. They are used to express relationships between different variables. In the case of inverse variation, we use the equation \(z = \frac{k}{t}\) to express how \(z\) changes relative to \(t\).

This equation arises from translating the phrase 'varies inversely' into mathematical terms. Here, as the variable \(t\) increases, \(z\) decreases, maintaining a product that equals the constant \(k\). This relationship is pivotal and forms the backbone of problems dealing with inverse variations.

Applying the Substitution Method

The substitution method is a powerful technique used to solve equations, especially when you aim to find an unknown. By substituting known values into an equation, we can isolate and solve for the unknown variable.

In the given problem, we have \(t = 3\) and \(z = 5\). By substituting these values into the inverse variation equation \(z = \frac{k}{t}\), we get \(5 = \frac{k}{3}\). By solving this equation, we get \(k = 15\). This process is straightforward and helps determine crucial constants like \(k\) in inverse variation problems.

Substitution not only makes solving equations manageable but also allows a clear path from problem to solution, reinforcing the connection between math theory and practice.