Problem 14 - Algebra and Trigonometry

Question

Write an equation that expresses the statement. \(S\) is jointly proportional to the squares of \(r\) and \(\theta\)

Step-by-Step Solution

Answer

The equation is \( S = k imes r^2 imes \theta^2 \).

Step 1: Understanding Joint Variation

The statement 'jointly proportional' means that a variable is proportional to the product of two or more variables. Here, 'S is jointly proportional to the squares of r and θ' indicates that S varies directly with the product of square of r and square of θ.

Step 2: Introducing Proportionality Constant

In a typical proportional relationship, a constant of proportionality, denoted as k, is used to transform the proportionality into an equation. This is because S is not necessarily the product of the squares of r and θ; instead, it is scaled by a constant factor.

Step 3: Setting Up the Proportional Equation

Based on the statement, we can write the equation: \( S = k imes r^2 imes heta^2 \), where S is the variable, and \( r^2 \) and \( \theta^2 \) are the squares of the independent variables.

Step 4: Writing the Final Equation

Thus, the equation expressing the joint variation of S with respect to the squares of r and θ is \( S = k imes r^2 imes \theta^2 \). This equation correctly represents S as jointly proportional to \( r^2 \) and \( \theta^2 \).

Key Concepts

Proportionality Constant

A proportionality constant is like a magic number in math equations. It bridges the relationship between two varying quantities. Imagine it as a knob that adjusts the strength of their connection. In our exercise, S doesn't just equal the product of \( r^2 \) and \( \theta^2 \); it’s scaled by this constant, represented as \( k \). Here’s why it's crucial:

• The proportionality constant allows for the equation to be fine-tuned to match real-world situations.
• It helps unify the measurement units between different quantities to ensure the equation holds true.

Think of \( k \) as the factor that scales or fits the relationship neatly. Without \( k \), the equation might not accurately portray how S behaves under different values of \( r \) and \( \theta \). Thus, \( k \) ensures our equation stays valid under various scenarios.

Product of Variables

The product of variables refers to the multiplication of variables—it’s like mixing elements to create a new entity. In mathematical terms, it involves multiplying variables or their powers. Here’s how it works in our context:

• When the statement mentions \( r^2 \) and \( \theta^2 \), it refers to multiplying the squares of \( r \) and \( \theta \).
• This multiplication shows how one variable relates to others linearly or non-linearly.

For S to vary jointly with \( r^2 \) and \( \theta^2 \), their product signifies a compounded effect. Simply put, changing \( r \) or \( \theta \) impacts S more significantly than if they were considered individually. Thus, the product of these squared variables illustrates how changes in each variable influence S altogether.

Direct Variation

Direct variation is a straightforward yet powerful relationship in mathematics. It describes a scenario where increasing one quantity results in a proportional increase in the other. This concept applies to situations where one variable changes uniformly with another. Let’s see how it’s connected to our case:

• Here, S varies directly with the product \( r^2 \theta^2 \).
• If either \( r \) or \( \theta \) increases, S will increase, assuming the rest of the variables stay the same.

Direct variation implies a consistent ratio or relationship. This means, as the squared terms of \( r \) and \( \theta \) change, S changes proportionally, dictated by \( k \). Understanding direct variation helps us predict how changes in these variables impact S under controlled conditions, giving us a clear model for prediction.