Problem 43 - College Algebra

Question

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

Step-by-Step Solution

Answer

Combined variation relates three or more variables, where one quantity varies directly as the second and inversely as the third. An example is the speed of an object lifted by a pulley system, varying directly with applied force and inversely with its own weight.

Step 1: Define Combined Variation

Combined variation is a relationship between three or more quantities where one quantity varies directly as a result of a second but inversely with respect to the third quantity. It combines direct and inverse variations. In mathematics, this relationship is expressed as \(y = k \cdot \frac{x}{z}\), where \(k\) is the constant of variation, \(x\) and \(z\) are variables.

Step 2: Providing an example

Imagine an object being lifted into the air by a pulley system. The speed at which the object rises (let's denote this as \(y\)) is directly proportional to the force applied (denote this as \(x\)) and inversely proportional to the weight of the object (denote this as \(z\)). As more force is applied, the object rises faster (direct variation). But as the object becomes heavier (increased weight), the speed decreases (inverse variation). Therefore, the lifting speed is subject to combined variation. Mathematically, this could be modeled by the equation \(y = k \cdot \frac{x}{z}\).