Problem 35 - Elementary Algebra

Question

Construct a mathematical model given the following. \(y\) varies jointly as \(x\) and \(z\) and inversely as the square of \(w,\) where \(y=\) 30 when \(x=8, z=3,\) and \(w=2\)

Step-by-Step Solution

Answer

The mathematical model is \(y = \frac{5xz}{w^2}\).

Step 1: Understand Joint and Inverse Variation

A joint variation means that one variable varies proportionally with the product of two or more other variables. Specifically, if \(y\) varies jointly as \(x\) and \(z\), \(y = kxz\) for some constant \(k\). An inverse variation means one variable varies inversely with another variable, or a power thereof. If \(y\) varies inversely as \(w^2\), \(y = \frac{k}{w^2}\). Combining these, we have \(y = \frac{kxz}{w^2}\).

Step 2: Insert Known Values to Find Constant

We know that when \(y=30\), \(x=8\), \(z=3\), and \(w=2\). Plug these values into the equation \(y = \frac{kxz}{w^2}\): \[30 = \frac{k \times 8 \times 3}{2^2}\] This simplifies to \[30 = \frac{24k}{4}\].

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

Continue simplifying the equation from Step 2: \[30 = 6k\] Now, solve for \(k\) by dividing both sides by 6: \[k = \frac{30}{6} = 5\].

Step 4: Write the General Equation

With \(k\) found to be 5, substitute back into the joint and inverse variation equation: \[y = \frac{5xz}{w^2}\]. This is the model that represents the given variation conditions.

Key Concepts

Joint Variation

Joint variation is a way to describe how a variable depends proportionally on the product of two or more other variables. This concept is common in physics and engineering, where certain quantities are related. For example, if we say a variable \( y \) varies jointly with \( x \) and \( z \), it means:

• \( y \) increases as \( x \) and/or \( z \) increase, and
• \( y \) decreases as \( x \) and/or \( z \) decrease.

A simple representation of joint variation is given by the equation \( y = kxz \), where \( k \) is a constant known as the constant of variation. This equation means \( y \) is directly proportional to the product of \( x \) and \( z \). The term "joint" indicates that two or more variables affect \( y \) together, unlike simple direct variation where one variable affects \( y \). Understanding joint variation helps in modeling complex relationships between variables.

Inverse Variation

Inverse variation expresses how a variable depends inversely on another variable. This means as one variable increases, the other one decreases. The concept works the opposite way compared to direct variation. If \( y \) varies inversely as \( w \), it follows:

• \( y = \frac{k}{w} \) for some constant \( k \).

A more complex form involves powers of a variable, such as when \( y \) varies inversely as \( w^2 \). This is represented by:

• \( y = \frac{k}{w^2} \).

Inverse variation is useful when changes in one quantity result in opposite changes in another, like speed and travel time. In the given problem, \( y \) varies inversely with the square of \( w \) (\( w^2 \)), emphasizing the quadratic nature of the influence \( w \) has on \( y \). By understanding inverse variation, one can predict the behavior of variables affected in reverse proportion.

Constant of Variation

The constant of variation, denoted as \( k \), is a crucial component in both joint and inverse variations. It quantifies the specific relationship between variables. Essentially, \( k \) determines the "strength" of the change one variable experiences concerning others.For joint variation, the formula used is \( y = kxz \). The constant \( k \) allows us to scale the relationship between \( y \) and the product of \( x \) and \( z \).In inverse variation, \( k \) lets us understand the opposition in variable behavior, like \( y = \frac{k}{w^2} \), where \( k \) adjusts how \( y \) changes relative to the power of another variable.To find \( k \), one must insert known values of variables into the variation equation and solve for \( k \). This was achieved in the original problem by substituting the given values of \( y, x, z, \) and \( w \) into the variation model and solving for \( k \). Using \( k \) ensures a precise mathematical model which accurately describes the interaction of variables.