Problem 15 - Algebra and Trigonometry

Question

Write an equation that expresses the statement. \(R\) is proportional to \(i\) and inversely proportional to \(P\) and \(t\)

Step-by-Step Solution

Answer

The equation is \( R = k \times \frac{i}{P \, t} \).

Step 1: Understand the Proportionality Statement

To start, let's identify how each variable is related as described in the statement. When a variable is proportional to another, it means the first can be expressed as the second multiplied by a constant. When a variable is inversely proportional to another, it means the first can be expressed as a constant divided by the second.

Step 2: Express Direct Proportionality

The statement given is that \(R\) is proportional to \(i\). We express this using the equation: \(R = k_1 \, i\) where \(k_1\) is a constant of proportionality.

Step 3: Express Inverse Proportionality

It is also given that \(R\) is inversely proportional to \(P\) and \(t\). We express this by adding these inversely proportional terms: \(R = \frac{k_2}{P}\) and \(R = \frac{k_3}{t}\).

Step 4: Combine Proportional Relationships

To combine both direct and inverse proportional relationships into a single equation, we can express \(R\) combining all terms involving proportionalities with a single constant: \(R = k \times \frac{i}{P \, t}\), where \(k\) is a constant that incorporates all individual constants.

Step 5: Final Equation

The equation that summarizes all relationships is: \[ R = k \, \frac{i}{P \, t} \] This indicates that \(R\) changes directly with \(i\) and inversely with both \(P\) and \(t\).

Key Concepts

Direct Proportionality

Direct proportionality is a relationship between two variables where one variable is a constant multiple of the other. It is expressed in a mathematical equation as \( y = kx \), where \( y \) and \( x \) are the variables, and \( k \) is the constant of proportionality. This means:

• When one variable increases, the other increases at a constant rate.
• When one variable decreases, the other decreases proportionally.

For instance, if \( R \) is directly proportional to \( i \), we can describe this relationship with the equation \( R = k_1 i \). Here, \( R \) increases as \( i \) increases, assuming the constant \( k_1 \) remains unchanged. This simple yet powerful idea helps us predict how changes in one factor affect another.

Inverse Proportionality

Inverse proportionality describes a situation where one variable increases as another decreases. This relationship is captured by the equation \( y = \frac{k}{x} \), meaning that \( y \) is inversely proportional to \( x \). Important characteristics include:

• As one variable increases, the other decreases, assuming the constant \( k \) stays the same.
• The product of the variables remains constant. That is, \( y \times x = k \).

In the context of the exercise, \( R \) is inversely proportional to both \( P \) and \( t \), which can be described by the equation \( R = k \times \frac{1}{P \cdot t} \). This indicates that an increase in either \( P \) or \( t \) will cause \( R \) to decrease, highlighting the reciprocal nature of these relationships.

Constants and Variables

Constants and variables form the backbone of algebraic equations, representing fixed values and changing quantities, respectively. In proportional relationships, they play distinct roles:

• **Variables**: These are the letters or symbols that can change value, like \( R, i, P, \) and \( t \). They capture the idea of measurement and computation, reflecting real-world quantities or unknowns.
• **Constants**: Constants like \( k \) or specific numbers remain unchanged throughout the problem. They make the mathematical relationships precise, ensuring proportional changes are accurately described.

In our scenario, the constant \( k \) in the equation \( R = k \frac{i}{P \, t} \) ensures the relationship between \( R \), \( i \), \( P \), and \( t \) remains consistent. By recognizing constant and variable roles, students can better grasp how mathematical models reflect changes in reality.