Problem 18 - Algebra and Trigonometry

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 the constant of proportionality \( k = 15 \).

Step 1: Understanding Inverse Variation

In inverse variation, when one variable increases, the other decreases. We express inverse variation as \( z = \frac{k}{t} \), where \( k \) is the constant of proportionality.

Step 2: Substitute Known Values

We know that when \( t = 3 \), \( z = 5 \). Substitute these values into the equation: \( 5 = \frac{k}{3} \).

Step 3: Solve for the Constant of Proportionality, \( k \)

Multiply both sides of the equation \( 5 = \frac{k}{3} \) by 3 to solve for \( k \). \[ 5 \times 3 = k \] So, \( k = 15 \).

Step 4: Write the Final Equation

Now that we've found \( k = 15 \), substitute back into the inverse variation equation to get \( z = \frac{15}{t} \).

Key Concepts

Constant of Proportionality

In inverse variation, the constant of proportionality, denoted by \( k \), plays a key role. It signifies a fixed relationship between the variables involved. When you observe that one variable varies inversely as another, it means their product remains constant.
For example, if variable \( z \) varies inversely as \( t \), the equation is given by \( z = \frac{k}{t} \). Here, \( k \) is the constant that does not change, regardless of the values of \( z \) and \( t \). In our example, the known values \( t = 3 \) and \( z = 5 \) help us find this constant.

• Input these values into the inverse variation equation: \( 5 = \frac{k}{3} \).
• By solving the equation, you find \( k = 15 \), which means \( z \cdot t = 15 \) is the unchanging product.

Understanding this constant is crucial as it allows all future predictions regarding \( z \) and \( t \) when one value is known.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. They usually involve variables and constants. In the exercise at hand, the equation \( z = \frac{k}{t} \) is an example of such an equation used to describe inverse variation.
These equations help us model and solve real-world problems by using known values to predict unknown ones. A typical step involves substituting values into the equation, simplifying the expressions, and solving for the unknown.

• In the example, you start with the equation \( 5 = \frac{k}{3} \).
• Mobilize algebraic skills to isolate \( k \), resulting in \( k = 15 \).

This process also illustrates the power of algebraic equations in finding solutions to seemingly complex relationships.

Variable Substitution

Variable substitution is a critical step in solving algebraic problems. Here, it involves replacing variables with actual known values to find unknown constants or variables. This method simplifies the understanding of relationships in equations, especially in inverse variations.
With inverse variation, substitution helps confirm the relationship between \( z \) and \( t \) by inserting known numbers into equations.

• In our scenario, you substitute \( t = 3 \) and \( z = 5 \) into \( z = \frac{k}{t} \) which simplifies to \( 5 = \frac{k}{3} \).
• This substitution is pivotal in unlocking the constant \( k \) by simplifying the equation to find \( k = 15 \).

Through these steps, variable substitution emerges as a powerful tool, turning abstract equations into concrete solutions.