Problem 21 - College Algebra

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x\) and \(z\) and inversely as \(w .\) When \(x=3\), \(z=5\), and \(w=6,\) then \(y=10\).

Step-by-Step Solution

Answer

The equation is \( y = 4 \frac{xz}{w} \).

Step 1: Understand the Joint and Inverse Variation

When a variable varies jointly with two other variables, it means it is directly proportional to the product of those variables. If a variable varies inversely with another, it means it is inversely proportional to that variable. Therefore, for your problem: \( y = k \frac{xz}{w} \), where \( k \) is the constant of variation.

Step 2: Substitute Given Values to Find k

Substitute the given values of \( x = 3 \), \( z = 5 \), \( w = 6 \), and \( y = 10 \) into the equation to find the constant \( k \):\[ 10 = k \frac{3 \cdot 5}{6} \]Simplifying the right side gives:\[ 10 = k \frac{15}{6} \]\[ 10 = k \cdot 2.5 \]

Step 3: Solve for Constant of Variation k

To solve for \( k \), divide both sides by 2.5:\[ k = \frac{10}{2.5} \]\[ k = 4 \].

Step 4: Write the Final Equation

Insert \( k = 4 \) back into the general equation to write the final relationship:\[ y = 4 \frac{xz}{w} \].

Key Concepts

Direct Variation

Direct variation occurs when one variable is proportional to another variable. In simpler terms, as one variable increases, the other variable increases at a constant rate. For example, if you earn \(10 for every hour you work, your earnings vary directly with the number of hours worked. This relationship can be represented by the equation: \[ y = kx \]Here:

• \( y \) is your earnings,
• \( x \) is the number of hours worked,
• \( k \) is the constant of variation, which is \)10 in this case.

In joint variation, as seen in the exercise, a variable can vary directly with the product of two other variables, such as \( y = kxz \). In this form, \( y \) increases as either \( x \) or \( z \) increases, provided the other variables remain constant.

Inverse Variation

Inverse variation is a type of relationship where one variable increases as another variable decreases. Here's how it works: unlike direct variation, in inverse variation when one variable goes up, the other one goes down. The equation for inverse variation is:\[ y = \frac{k}{x} \]In this setup:

• \( y \) is the output variable.
• \( x \) is the input variable.
• \( k \) is the constant of variation.

An example might be your speed and travel time: as your speed (\( x \)) increases, your travel time (\( y \)) decreases. The constant \( k \) remains unchanged regardless of \( x \) or \( y \) values. In the exercise, this is used in combination with joint variation: \( y = k \frac{xz}{w} \). This means \( y \) varies jointly with \( x \) and \( z \), and inversely with \( w \).

Constant of Variation

The constant of variation, often denoted as \( k \), serves as the glue that binds variables together in both direct and inverse variation equations. It's a crucial part of understanding how variables are related. For example, if \( y \) varies directly as \( x \), we have the equation:\[ y = kx \]Here, \( k \) is the constant that shows the ratio of \( y \) to \( x \). When determining \( k \), you're essentially figuring out how much one variable will change in relation to a unit change in another variable. In joint variation, \( k \) specifies how much the dependent variable changes with respect to the products of changes in other variables as in \( y = kxz \). Meanwhile, in the inverse context, \( k \) becomes the factor representing their inverse relationship like in \( y = \frac{k}{x} \). In the provided solution, the constant \( k = 4 \) was found by substituting known values into the joint and inverse variation equation.

Proportional Relationships

Proportional relationships describe scenarios where two quantities maintain a constant ratio. In simpler terms, they are situations where if one number doubles, the other doubles as well, keeping the same proportion between them. These relationships fall under both direct and inverse variation, depending on how the variables interact.

• In direct proportionality, the ratio is simply the constant of variation. Both variables increase or decrease together. This can be seen in a straight-line graph passing through the origin.
• In inverse proportionality, as one variable increases, the other decreases in such a way that the product of the two variables remains constant. This creates a hyperbolic graph.

Joint and inverse variations like \( y = k \frac{xz}{w} \) blend these concepts by forming a relationship where \( y \) varies jointly with \( x \) and \( z \) and inversely with \( w \). Each part of this equation showcases a proportional relationship resulting in a balanced equation that represents the given condition.