Problem 53 - College Algebra

Question

Explain what is meant by combined variation. Give an example with your explanation.

Step-by-Step Solution

Answer

Combined variation is when a quantity varies directly with a certain quantity and inversely with another. An example may be an equation such as \(z = ky/x\), where \(z\) varies directly with \(y\) and inversely with \(x\). It's a combination of direct and inverse variation.

Step 1: Understand Direct and Inverse Variation

To clarify the concept of combined variation, an understanding of direct and inverse variation is crucial. In direct variation, an increase or decrease in one quantity results in a proportional increase or decrease in the other quantity. In inverse variation, an increase in one quantity results in the decrease of the other quantity and vice versa, maintaining that the product is constant. For instance, if we denote two variables as \(x\) and \(y\), in direct variation, we can write the equation as \(y = kx\), where \(k\) is the constant of variation. In inverse variation, the equation takes the shape of \(yx = k\) or \(y = k/x\).

Step 2: Define Combined Variation

A combined variation is a relationship where a quantity varies directly as the product or quotient of two or more other quantities. It essentially combines the concepts of direct and inverse variation.

Step 3: Example of Combined Variation

An example of combined variation can be an equation like \(z = ky/x\) where z varies directly with \(y\) and inversely with \(x\). This means if \(y\) increases, \(z\) increases and if \(x\) increases, \(z\) decreases provided \(k\) remains constant. This is a classic case of combined variation.