Problem 44 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Joint variation is when a quantity varies directly or inversely with more than one other quantity. An example of direct joint variation is the Ideal Gas Law: \(V = kTn\), where the volume \(V\) of a gas varies directly with its temperature \(T\) and the number of moles \(n\). An example of inverse joint variation is the Law of Gravitation: \(F = G*m1*m2/(r^2)\), where the force \(F\) varies inversely with the square of the distance \(r\) between the two objects and directly with the product of their masses \((m1*m2)\).

Step 1: Definition of Joint Variation

Joint variation (or combined variation) is a type of variation where a quantity varies directly or inversely with more than one other quantity. It's often expressed with the following form: \(z = kxy\) if z varies directly with x and y, or \(z = k/(xy)\) if z varies inversely with x and y, where k is the constant of variation.

Step 2: Example of Direct Joint Variation

For instance, consider an example where the volume \(V\) of a gas varies directly with its temperature \(T\) and the number of moles \(n\) according to the Ideal Gas Law: \(V = kTn\). Here, k is a constant. If the temperature and number of moles both double, the volume of the gas also doubles.

Step 3: Example of Inverse Joint Variation

An example of inverse joint variation can be the physical law of gravitation, where the force \(F\) varies inversely with the square of the distance \(r\) between the two objects and directly with the product of their masses \((m1*m2)\). This relationship is given by the equation: \(F = G*m1*m2/(r^2)\), where G is the constant of variation and measures the gravity force.