Problem 11 - Algebra and Trigonometry

Question

1–12 ? 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 \( R = k \frac{i}{Pt} \).

Step 1: Understand Proportional Relationships

The statement tells us that a variable, \( R \), has two types of proportional relationships. It's directly proportional to \( i \) and inversely proportional to both \( P \), and \( t \). This means that as \( i \) increases, \( R \) also increases, but as \( P \) or \( t \) increases, \( R \) decreases.

Step 2: Setting Up the Proportional Equation

To express the relationships mathematically, we introduce a constant of proportionality, \( k \), to write the equation. The equation will look like this: \( R = k \frac{i}{Pt} \). This equation captures both the direct and inverse proportionalities.

Step 3: Define Variables and Constant

In this equation: \( R \) is the variable we are solving for, \( i \) is directly proportional, \( P \) and \( t \) are inversely proportional. \( k \) is a constant that represents the proportionality factor, which must be determined from given conditions.

Step 4: Final Solution

The equation that expresses the given statement is: \( R = k \frac{i}{Pt} \). This equation shows how \( R \) changes in relation to changes in \( i \), \( P \), and \( t \).

Key Concepts

Direct Proportionality

Direct proportionality refers to a straightforward relationship between two variables, where one variable increases or decreases in direct response to the other. If one variable doubles, the other variable also doubles.

This relationship can be expressed with the formula: \[ y = kx \]where:

• \( y \) and \( x \) are the two directly proportional quantities
• \( k \) is the constant of proportionality

The constant \( k \) remains unchanged despite variations in \( x \).

Direct proportionality is like the relation between distance traveled and time at a constant speed. If time doubles, the distance also doubles, provided the speed remains constant. This predictable relationship simplifies calculations and is frequently seen in science and economics.

Inverse Proportionality

Inverse proportionality describes a relationship where one quantity increases while the other decreases. This opposite behavior signifies that the variables multiply to a constant value.

The equation for inverse proportionality is:\[ x \cdot y = k \]where:

• \( x \) and \( y \) are the two inversely proportional quantities
• \( k \) is the constant product of \( x \) and \( y \)

Unlike direct proportionality, increasing one factor causes the other to decrease, maintaining a constant product.

Consider speed and travel time for a fixed distance. If the speed increases, the time to cover that distance decreases correspondingly, hence they are inversely proportional. Recognizing this helps in optimizing processes and understanding constraints in various situations.

Constant of Proportionality

The constant of proportionality, \( k \), is a critical element in defining the relationship between proportional variables. It represents the unchanging factor in equations for both direct and inverse proportionality.

In a direct proportionality equation:\[ y = kx \]the constant \( k \) is the ratio of \( y \) to \( x \). It signifies how much \( y \) changes with respect to \( x \). Inverse proportionality involves:\[ x \cdot y = k \]here, \( k \) is the product that stays fixed despite the changing values of \( x \) and \( y \).

• It helps understand the scale of variation in relationships
• Provides a measure to predict changes

Knowing \( k \) in practical scenarios allows predictions and adjustments in fields like physics, economics, and engineering. It keeps calculations consistent and comparable across various contexts.