Complex Population Growth and Decay Models

Propotionality

The variable y is directly proportional to the variable x with proportionality constant ~0.6.

The variable y is inversely proportional to the variable x with proportionality constant 1.

In mathematics, two varying quantities are said to be in a relation of proportionality, multiplicatively connected to a constant; that is, when either their ratio or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant.

If the ratio ( ${\frac {y}{x}}$ ) of two variables ( $x$ and $y$ ) is equal to a constant (k = ${\frac {y}{x}}$ ), then the variable in the numerator of the ratio ( $y$ ) can be product of the other variable and the constant $(y = k \cdot x)$ . In this case $y$ is said to be directly proportional to $x$ with proportionality constant $k$ . Equivalently one may write $x={\frac {1}{k}}$ ⋅ y; that is, $x$ is directly proportional to $y$ with proportionality constant ${\frac {1}{k}}=({\frac {x}{y}})$ . If the term proportional is connected to two variables without further qualification, generally direct proportionality can be assumed.
If the product of two variables $(x \cdot y)$ is equal to a constant $(k = x \cdot y)$ , then the two are said to be inversely proportional to each other with the proportionality constant $k$ . Equivalently, both variables are directly proportional to the reciprocal of the respective other with proportionality constant $k$ ( $x$ = k ⋅ ${\frac {1}{y}}$ and $y = k \cdot$ ${\frac {1}{x}}$ ).

If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., ${\frac {a}{b}}={\frac {x}{y}}$ = ⋯ = k (for details see Ratio).

Direct proportionality

Given two variables x and y, y is directly proportional to x if there is a non-zero constant k such that

y=kx.

The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter alpha) or "~":

y\propto x,

or

y\sim x.

For $x\neq 0$ the proportionality constant can be expressed as the ratio

k={\frac {y}{x}}.

It is also called the constant of variation or constant of proportionality.

A direct proportionality can also be viewed as a linear equation in two variables with a y-intercept of 0 and a slope of k. This corresponds to linear growth.

Examples

If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
On a map of a sufficiently small geographical area, drawn to scale distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map.
The force, acting on a small object with small mass by a nearby large extended mass due to gravity, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as gravitational acceleration.
The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, Newton's second law, is the classical mass of the object.

Solve problems involving joint variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c, cost, varies jointly with the number of students, n, and the distance, d.

A GENERAL NOTE: JOINT VARIATION

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if x varies directly with both y and z, we have $x=kyz$ . If x varies directly with y and inversely with z, we have $x={\frac {ky}{z}}$ .

Notice that we only use one constant in a joint variation equation.

Example

A quantity x varies directly with the square of y and inversely with the cube root of z. If x = 6 when y = 2 and z = 8, find x when y = 1 and z = 27.

Solution

Begin by writing an equation to show the relationship between the variables.

$x={\frac {k{y}^{2}}{\sqrt[{3}]{z}}}$

Substitute x = 6, y = 2, and z = 8 to find the value of the constant k.

${\begin{cases}6={\frac {k{2}^{2}}{\sqrt[{3}]{8}}}\\6={\frac {4k}{2}}\\3=k\end{cases}}$

Now we can substitute the value of the constant into the equation for the relationship.

$x={\frac {3{y}^{2}}{\sqrt[{3}]{z}}}$

To find x when y = 1 and z = 27, we will substitute values for y and z into our equation.

${\begin{cases}x={\frac {3{\left(1\right)}^{2}}{\sqrt[{3}]{27}}}\\{\text{ }}=1\end{cases}}$