Problem 16 - Algebra and Trigonometry

Question

13–22 ? 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 constant of proportionality is 9; the equation is \( S = 9pq \).

Step 1: Understand Joint Variation

When a variable varies jointly as two other variables, it means that the variable is proportional to the product of those two variables. Thus, if S varies jointly as p and q, we express it as \( S = kpq \), where \( k \) is the constant of proportionality.

Step 2: Substitute Known Values

We know from the problem that when \( p = 4 \) and \( q = 5 \), \( S = 180 \). Substitute these values into the equation: \( 180 = k \times 4 \times 5 \).

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

In the equation \( 180 = k \times 4 \times 5 \), simplify and solve for \( k \):\( 180 = 20k \).Divide both sides by 20:\( k = \frac{180}{20} \).Calculate:\( k = 9 \).

Step 4: Write the Final Equation

With the constant of proportionality \( k = 9 \), the equation expressing the joint variation of \( S \) as \( p \) and \( q \) is \( S = 9pq \).

Key Concepts

Constant of Proportionality

The constant of proportionality is a key element in describing how quantities are connected in proportional relationships. When we talk about variation, whether it is direct, inverse, or joint, like in this exercise, the constant of proportionality helps us understand how one variable affects another. The constant is typically denoted by the letter "k."

• In joint variation, if a variable like \( S \) varies jointly as \( p \) and \( q \), the equation is \( S = kpq \).
• Here, \( k \) tells us how strongly \( S \) is affected by changes in \( p \) and \( q \) together.
• The constant remains the same for a particular relationship, meaning it doesn't change even if the values of \( p \) or \( q \) change, as long as the relationship holds.

Finding \( k \) involves substituting known values of \( p \), \( q \), and \( S \) into the equation. This calculation is crucial because it allows us to predict new values of \( S \) for different \( p \) and \( q \) values.

Algebraic Equations

Algebraic equations are the tools we use to solve problems involving unknowns. They serve as the canvas where relationships between different quantities are expressed. In the context of joint variation, algebraic equations convey how a variable depends on two others. Let's break down the process:

• Start by forming the base equation based on the variation type. For joint variation, it is \( S = kpq \).
• Substitute given real values for the variables to find the constant of proportionality \( k \).
• Solve the resulting equation to isolate \( k \), providing you with a full picture of the proportional relationship.

Algebraic manipulation, like simplifying and rearranging equations, is pivotal in finding \( k \). Mastering these skills not only helps in solving math exercises but also enhances problem-solving abilities in real-world scenarios.

Proportional Relationships

Proportional relationships are foundational concepts in mathematics that describe how variables relate to each other. In a proportional relationship, two quantities increase or decrease at the same rate. Joint variation is a special type where a variable is influenced by the product of two other variables. Let's see how it applies:

• When \( S \) varies jointly with \( p \) and \( q \), it means changes in \( p \) and \( q \) simultaneously affect \( S \).
• This relationship is captured in the equation \( S = kpq \), where \( k \) is found using known values of \( p \), \( q \), and \( S \).
• Once \( k \) is known, the equation can predict \( S \) for any other values of \( p \) and \( q \) that adhere to the same proportional relationship.

Understanding proportional relationships allows us to anticipate how a change in one part of a system influences others. This has applications across many fields, from physics to economics, where predicting outcomes based on known variables is essential.