Problem 77 - Elementary Algebra

Question

Construct a mathematical model given the following. $$ y \text { is jointly proportional to } x \text { and } z \text { , where } y=36 \text { when } x=3 \text { and } z=4 \text { . } $$

Step-by-Step Solution

Answer

The model is \( y = 3xz \).

Step 1: Understand Joint Proportionality

Joint proportionality means that one variable, say \( y \), is proportional to the product of two other variables, \( x \) and \( z \). Mathematically, this is expressed as \( y = kxz \), where \( k \) is the constant of proportionality.

Step 2: Substitute Known Values

We know that when \( y = 36 \), \( x = 3 \), and \( z = 4 \). Substitute these values into the equation \( y = kxz \):\[ 36 = k \times 3 \times 4 \]

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

Simplify the expression from the previous step:\[ 36 = 12k \]Next, solve for \( k \) by dividing both sides by 12:\[ k = \frac{36}{12} = 3 \]

Step 4: Write the Mathematical Model

Now that we have determined \( k = 3 \), substitute \( k \) back into the joint proportionality equation. The mathematical model is:\[ y = 3xz \]

Key Concepts

Constant of Proportionality

The constant of proportionality, often denoted as \( k \), plays a vital role in equations that express proportional relationships. In the context of joint proportionality, this constant links one variable to the product of two others. In simpler terms, if \( y \) is jointly proportional to \( x \) and \( z \), it means that their relationship can be described by the equation \( y = kxz \). The constant \( k \) ensures that this equation holds true given specific values.

To determine \( k \), you take known values of \( y \), \( x \), and \( z \) and plug them into the equation. For example, if \( y = 36 \), \( x = 3 \), and \( z = 4 \), you would substitute these into \( y = kxz \) to find \( k \). This involves simple algebraic operations:

• Start by replacing \( y \), \( x \), and \( z \) with their known values.
• Simplify the equation to isolate \( k \).
• The result is your constant of proportionality.

This constant is a fixed value that uniquely defines the relationship given the context of the problem.

Mathematical Modeling

Mathematical modeling is the process of representing a real-world scenario with mathematical expressions. In joint proportionality problems, this involves creating an equation that describes how one quantity varies with others. For instance, if you know that a quantity \( y \) is jointly proportional to \( x \) and \( z \), your goal is to construct the model \( y = kxz \).

This model provides a structured way to understand and predict behaviors of the quantities involved when \( x \) and \( z \) change. To articulate a precise model, follow these steps:

• Identify the type of proportionality: Here it's joint proportionality.
• Determine the relation and form the equation \( y = kxz \).
• Find the constant \( k \) using given data points.

Once the model is established, you can use it to extrapolate or predict other scenarios by plugging in different values for \( x \) and \( z \). This makes mathematical modeling an invaluable tool in understanding complex systems in a simplified manner.

Solving Equations

Solving equations accurately is a fundamental skill in mathematics, especially when dealing with proportions. In joint proportionality problems, you often need to solve an equation like \( 36 = k \times 3 \times 4 \) to find the constant of proportionality. This requires some basic arithmetic steps:

First, simplify the right side of the equation by multiplying the numbers. For example, \( 3 \times 4 = 12 \).
Next, rearrange the equation to isolate \( k \) on one side by performing operations like division or multiplication:

• If \( 36 = 12k \), divide both sides by 12.
• This simplifies the expression, resulting in \( k = \frac{36}{12} \).
• The solution gives \( k = 3 \), completing the process.

By thoroughly understanding these steps, you can approach similar problems with confidence. The act of solving equations doesn't just find the unknown values; it also confirms that your proportional relationships are correctly represented in your model.