Often the value of a variable quantity depends on the values of several other quantities, for instance the amount of simple interest on an investment depends on the interest rate, the amount invested, and the period of time involved. For compound interest the amount of interest depends on an additional variable: how often the compounding takes place. A situation in which one variable depends on several others is called combined variation. An important type of combined variation is defined below.
If a variable quantity is proportional to the product of two or more variable quantities we say that is jointly proportional to these quantities, or varies jointly as (or with) these quantities.
For instance, if is jointly proportional to and , then is related to and by the formula
where is a constant. Sometimes the word “jointly” is omitted and we simply say that is proportional to and .
In chemistry, the absolute temperature of a perfect gas varies jointly as its pressure and its volume . Given that Kelvin when pounds per square inch and cubic inches, find a formula for in terms of and find when pounds per square inch and cubic inches.
Since varies jointly as and , there is a constant such that . Putting , and , we find that or
Thus, the desired formula is
When and , we have Kelvin.
We sometimes have joint variation together with inverse variation. Perhaps the most important historical discovery of this type of combined variation is Newton’s law of universal gravitation, which states that the gravitation force of attraction between two particles is jointly proportional to their masses and , and inversely proportional to the square of the distance between them. In other words, is related to , and by the formula
where is constant of proportionality.
Careful measurements show that two kilogram masses meter apart exact a mutual gravitational attraction of Newton (one pound of force is approximately Newton). The Earth has a mass of kilograms. Find the Earth’s gravitational force on a space capsule that has a mass of kilograms and that is meters from the center of the Earth.
Putting Newton, kilogram, kilogram, and meter in the formula , we find that
So that, for arbitrary values of , and ,
Now we substitute ,, and to obtain
(or less than pounds of gravitational force).