Problem 22 - Precalculus Mathematics for Calculus

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. \(S\) varies jointly as \(p\) and \(q .\) If \(p=4\) and \(q=5,\) then \(S=180\).

Step-by-Step Solution

Answer

The equation is \(S = 9pq\). The constant of proportionality is 9.

Step 1: Understand the Joint Variation

When a quantity varies jointly as two other quantities, it implies that it is directly proportional to the product of those two quantities. Hence, the relationship can be expressed as \(S = k imes p imes q\), where \(k\) is the constant of proportionality.

Step 2: Substitute Known Values

Substitute the given values into the equation to find the constant \(k\). We know that when \(p = 4\) and \(q = 5\), then \(S = 180\). So, substitute these values into \(S = k \times p \times q\): \(180 = k \times 4 \times 5\).

Step 3: Solve for the Constant of Proportionality

To find \(k\), isolate \(k\) by dividing both sides of the equation by \(20\) (since \(4 \times 5 = 20\)). Therefore, \(k = \frac{180}{20} = 9\).

Step 4: Write the Final Equation with the Constant

Now that we have found the constant of proportionality, the algebraic expression of the statement is \(S = 9pq\).

Key Concepts

Constant of Proportionality

When one variable varies in relation to another variable, a constant factor is often involved, which is known as the constant of proportionality. In the case of joint variation, such as the problem we are analyzing, this constant connects three variables. Joint variation means that one variable depends directly on the product of two other variables. Here, when we say that "\( S \) varies jointly as \( p \) and \( q \)," it means that \( S \) is directly proportional to the product of \( p \) and \( q \).
So, we express this relationship mathematically as:

• \( S = k \cdot p \cdot q \)

where \( k \) is the constant of proportionality. The constant allows us to calculate one variable if the others are known. When you substitute known values for \( p \), \( q \), and \( S \), you can then solve for \( k \). This constant provides consistency across different scenarios, ensuring that the relationship between these variables remains the same regardless of the specific numbers involved.

Direct Variation

Direct variation is a concept where one variable changes directly as another variable changes. In the case of joint variation, the direct variation extends this relationship to multiple variables. So, instead of relying solely on one variable, direct variation in joint variation depends on the product of two or more factors.
To break it down: if \( S \) directly varies with \( p \) and \( q \), then any increase in \( p \) or \( q \) will lead to a proportional increase in \( S \), provided that \( k \), the constant of proportionality, remains unchanged. This demonstrates a consistent and predictable relationship between the variables:

• When \( p \) increases, \( S \) increases.
• When \( q \) increases, \( S \) increases.
• If \( k \), \( p \), and \( q \) are known, \( S \) can be accurately calculated.

This makes it incredibly useful for understanding complex systems where multiple factors influence an outcome.

Algebraic Expression

An algebraic expression is a mathematical phrase that can incorporate numbers, variables, and operators, yet doesn’t equate to anything by itself. It's a way to represent mathematical ideas using symbols and letters. In the given exercise, we develop an algebraic expression to represent a joint variation scenario. Initially, the expression takes the form \( S = k \cdot p \cdot q \).This algebraic expression is pivotal because it encapsulates the entire relationship between the variables \( S \), \( p \), and \( q \) with the constant \( k \).

• \( S \): Dependent variable that changes based on \( p \) and \( q \).
• \( k \): Constant that locks the relationship consistent across various situations.
• \( p \) and \( q \): Independent variables whose product determines \( S \).

After finding the constant of proportionality \( k \), the completed expression becomes \( S = 9pq \). This expression provides all the information needed to solve problems involving \( S \), \( p \), and \( q \) in the context of this exercise.