Problem 21 - Intermediate Algebra

Question

Find the constant of variation for each of the stated conditions. \(y\) varies directly as \(x\) and inversely as \(z\), and \(y=45\) when \(x=18\) and \(z=2\).

Step-by-Step Solution

Answer

The constant of variation is 5.

Step 1: Understand the Variation Relationship

The problem states that \(y\) varies directly with \(x\) and inversely with \(z\). This can be expressed with the formula \(y = k \frac{x}{z}\), where \(k\) is the constant of variation.

Step 2: Substitute the Given Values

We know \(y=45\), \(x=18\), and \(z=2\). Substitute these values into the variation equation:\[45 = k \frac{18}{2}\].

Step 3: Simplify the Equation

Simplify the fraction on the right side of the equation: \(\frac{18}{2} = 9\). The equation becomes:\[45 = 9k\].

Step 4: Solve for the Constant of Variation, \(k\)

To find \(k\), divide both sides of the equation by 9:\[k = \frac{45}{9}\].Simplify the fraction: \(k = 5\).

Key Concepts

Direct Variation

Direct variation occurs when two variables change in the same proportion. Specifically, if one variable increases, the other variable increases at the same rate. The relationship between these variables can be expressed as \( y = kx \), where \( k \) is the constant of variation. This constant represents the ratio of \( y \) to \( x \).

You can think of direct variation like the relationship between distance and speed. If you travel faster, the distance you cover also increases proportionally. For example, if you double your speed, the distance covered in the same time doubles as well.

• Straight-line relationships: The graph of a direct variation is a straight line that passes through the origin.
• Proportional change: As one variable doubles, the other doubles too.

Understanding this concept helps in simplifying and solving equations effectively by determining the proportional relationships between variables.

Inverse Variation

Inverse variation is a relationship between two variables where one variable increases as the other decreases. The product of the two variables always remains constant. It can be expressed using the formula \( y = \frac{k}{z} \), where \( k \) is the constant of variation.

Imagine a seesaw: if one side goes up, the other must come down to keep balance. This is similar to how inverse variation works—the product of certain changing variables remains constant, maintaining balance.

• Inverse proportion: As one variable doubles, the other must halve to maintain equality.
• Graph interpretation: Inverse variation graphs are hyperbolas. Curves that decrease as they approach the axes.

Inverse variation is useful in situations where the increase of one aspect results in the decrease of another, yet the total effect or outcome remains consistent.

Variation Equations

Variation equations combine elements of both direct and inverse variations to show how changes in one or more variables affect another variable. These equations help clarify these relationships by using a blend of both variation types.

In our exercise, the variation equation is given as \( y = k \frac{x}{z} \). This illustrates that \( y \) directly varies with \( x \) (meaning \( y \) increases as \( x \) increases) and inversely varies with \( z \) (meaning \( y \) decreases as \( z \) increases).

• Application of multiple concepts: Variation equations often combine expectations from direct and inverse relationships.
• Solving Variation Equations: Identify known values and solve for the unknown variable, typically the constant \( k \).

Understanding variation equations is essential in modeling real-world situations, such as physics problems or economic models, where changes in multiple factors need to be considered simultaneously.