Problem 44 - College Algebra

Question

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

Step-by-Step Solution

Answer

Joint variation is a relationship between three or more quantities such that the quantity of one varies not alone but jointly as a product of two or more other quantities. An example of joint variation is the volume of a rectangular prism which varies jointly with its length, width, and height, expressed as \(V = l \cdot w \cdot h\).

Step 1: Understanding Joint Variation

Joint variation, or combined variation, is a type of relationship between variables where the dependent variable varies directly with the product of two or more other variables. This can be expressed with the formula \(y = k \cdot xz\), where \(y\) is the varying quantity, \(x\) and \(z\) are independent variables that \(y\) jointly varies with, and \(k\) is the constant of variation.

Step 2: Giving an Example

To illustrate this, consider the example of a rectangular prism. The volume of the prism varies jointly with its length, width, and height. If we let \(V\) represent the volume, \(l\) the length, \(w\) the width, and \(h\) the height, we can write the joint variation as \(V = l \cdot w \cdot h\). Here, the volume of the prism will change when any of the dimensions (length, width, or height) change, but the relationship between these four quantities always holds true.