In the later years of his life, the Italian scientist Galileo Galilei (1564 – 1642) wrote about his experiments with motion in a treatise called Dialogues Concerning Two New Sciences. Here he described his wonderful discovery that distances covered in consecutive equal time intervals by balls rolling down inclined planes are proportional to the successive odd positive integers. Thus, if denotes the distance covered during the time interval, then .
Here, is a suitable constant? Galileo determined that the constant depends only on the incline and not on the mass of the ball or the material which it is composed of. He reasoned that the same result should hold for freely falling bodies if air resistance is neglected.
Galileo’s discovery of a mathematical model for uniformly accelerated motion is considered to be the beginning of the science of dynamics. A mathematical model is an equation or a set of equations in which letters represent real world quantities, for instance, distance and time intervals in the equation . A mathematical model is often an idealization of the real world situation it supposedly describes. For instance, Galileo’s mathematical model assumes a perfectly smooth inclined plane, no friction, and no air resistance, and it neglects the rotational movement of the inertia of the balls as they roll down the inclined plane.
We study mathematical models related to the concept of variation.
Thus, the letters and in the following definition could represent variable quantities in a mathematical model.
If there is a constant such that
holds for all values of , we say that is directly proportional to or that varies directly as (or with ). The constant is called the constant of proportionality or the constant of variation.
For instance, if an automobile is moving at a constant speed of miles per hour, then the distance it travels in hours is given by the formula
Here and are variables in the mathematical model describing the motion of the automobile; the distance is directly proportional to the time , and the constant of proportionality is miles per hour.
Suppose that and are variables and that
so that is directly proportional to with as the constant of proportionality.
Note that when . Also, if , we have
So the fraction, or ratio, maintains the constant value as varies through nonzero values. Thus, if a nonzero value of and the corresponding value of are known, the value of can be determined.
Suppose that the rate at which impulses are transmitted along a nerve fiber is directly proportional to the diameter of the fiber. Given that meters per second when micrometers:
(a) Write a formula that express in terms of .
(b) Find if micrometers.
Since is directly proportional to , we have
where is the constant of proportionality.
(a) For , we have
Substituting meters per second and micrometers into this equation, we find that . Therefore, for all values of , we have .
(b) When micrometers, we have meters per second.
If we say that varies directly as the square of , we mean that and are related by an equation of the form
for some constant of variation . For instance, the formula for the area of a circle states that is directly proportional to the square of the radius , and that is the constant of variation. More generally:
Variation with the nth power
If there are constants and such that
This holds for all values of , so we say that is directly proportional to the nth power of or that varies directly as (or with) the nth power of .
The formula for the volume of a sphere states that is directly proportional to the cube of the radius and the constant of variation is . Find the volume of a sphere of radius centimeters and round off your answer appropriately.
Assuming that the radius centimeters is correct to three significant digits, we have rounded off our answer to three significant digits.
The situation in which is proportional to the reciprocal of arises so frequently that it is given a special name: inverse variation.
If there is a constant such that
For all nonzero values of , we say that is inversely proportional to or that varies inversely as (or with ).
Naturally, if for some constant and some positive rational number , we say that is inversely proportional to the nth power of .
A company is informed by a shopkeeper that the number of units of a certain item sold per month seems to be inversely proportional to the quantity , where is the selling price per unit in dollars. Suppose that units per month are sold when the price is per unit. How many units per month would be sold if its price were dropped to per unit?
Since is inversely proportional to , there is a constant such that
We know that when , so we have
Therefore, the formula
gives in terms of . For dollars, we have
So units per month would be sold at a price of 5$$ per unit.