Problem 22 - Algebra and Trigonometry

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 constant of proportionality is 9.

Step 1: Understand Joint Variation

When a variable varies jointly with two other variables, it means it is directly proportional to the product of these variables. We can express this as an equation: \[ S = k \cdot p \cdot q \] where \( k \) is the constant of proportionality.

Step 2: Substitute Given Values

Use the values provided in the problem to substitute into the equation: \[ 180 = k \cdot 4 \cdot 5 \] This will allow us to solve for the constant of proportionality \( k \).

Step 3: Simplify the Equation

Simplify the right side of the equation by multiplying:\[ 180 = 20k \] This is because \( 4 \cdot 5 = 20 \).

Step 4: Solve for the Constant of Proportionality

Divide both sides of the equation by 20 to solve for \( k \): \[ k = \frac{180}{20} = 9 \] Thus, the constant of proportionality \( k \) is 9.

Key Concepts

Proportionality Constant

In mathematics, the proportionality constant is a fixed number that relates two variables which vary together according to a specific relationship. In our context, when a variable varies jointly, it is proportional to the product of two other variables. This gives us an equation where a constant, called the proportionality constant, comes into play.
For example, in the exercise, the proportionality constant is represented by the letter \( k \). It acts as a multiplier that scales how the product of \( p \) and \( q \) affects \( S \), the main variable.
When figuring out the proportionality constant, we substitute given values into the equation. This helps isolate \( k \), allowing us to calculate its value. In our exercise, using the values given \( S = 180 \), \( p = 4 \), and \( q = 5 \), we derived \( k = 9 \). This tells us that every time the product of \( p \) and \( q \) influences \( S \), it does so through multiplication by 9.

Direct Variation

Direct variation occurs when two variables change in proportion with each other, meaning if one increases, the other increases by a related factor. In joint variation, direct variation principles apply to more than two variables, such as \( S \) varying jointly with \( p \) and \( q \).
Joint variation can seem complex but can be broken down using equations where one variable changes directly in relation to multiple others. It’s a neat way to understand how changes in one set of variables can cause changes in another. For example, if either \( p \) or \( q \) increases, \( S \) will also increase, as long as the other values remain constant.

• Understanding direct variation helps explain the relationship between variables easily.
• Remember, more than just two variables can participate in joint variation, unlike typical direct variation.

This kind of variation is useful in modeling real-world situations, such as physics where forces, distances, and times proportionally interact.

Algebraic Equations

Algebraic equations form the basis for expressing mathematical relationships like joint variation. These equations use symbols and formulas to represent real-world problems.
In the exercise, we used an algebraic equation \( S = k \cdot p \cdot q \), to express joint variation, showing how \( S \) relates to \( p \) and \( q \) through \( k \), the proportionality constant.
Solving algebraic equations often involves:

• Substituting known values to simplify equations
• Isolating the variable you need to solve for, which in this case was \( k \)
• Using arithmetic operations like multiplication and division

These steps help reveal unknowns, showing how interconnected variables relate to each other. As seen in this problem, solving algebraic equations is crucial in applying mathematical principles to practical scenarios.