Many mathematical models are constructed using the principle of proportionality.
The basic idea is to identify two quantities that are expected to covary in
a proportional manner and write an equation capturing that relationship.
For some models, the proportionality constant is left unspecified as a parameter.
In other cases, the constant is either known or can be computed from the given
information. This subsection contains a variety of examples illustrating the power
of using proportional models.

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Subsubsection7.1.3.1Per Capita Rates

In population biology, we are often interested the relationship between
vital rates, such as birth rates and death rates, with the size of the population.
That is, we might expect that the birth rate (number of births per unit time)
would be larger if the population were larger since there are more individuals
potentially giving birth. Similarly, we might expect that the death rate (number
of deaths per unit time) in a population would also be larger in a larger population.
Models in population biology are often constructed by describing these relations.

The simplest model capturing the idea that the birth rate is larger when the
population is larger uses an assumption that the birth rate is proportional
to the population size. Using variables, let \(B\) represent the birth rate
for a population and let \(P\) represent the size of the population. These
are covarying variables. Our assumption is that \(B \propto P\), meaning that
we expect that under all conditions, the ratio \(B : P\)
is always the same constant.

Using a lowercase \(b\) for this constant, our model is characterized by the relation
\begin{equation}B = b \cdot P. \label{men-4}\tag{7.1.1}\end{equation}
This proportionality constant has a special name, the *per capita birth rate*.
Here, I am speaking about the role of the constant, not the particular letter \(b\).
The phrase *per capita* has a literal translation of *per head*.
Because \(b\) is defined by the ratio \(\displaystyle b=\frac{B}{P}\),
in the case that \(P=1\) (population size is 1), \(b\) would equal \(B\),
the number of births per unit time of a single individual.

In a similar way, if we assume that the death rate of a population is proportional
to the size of the population, then we should consider the per capita death rate
\(d\) which leads to a model relation between the population death rate \(D\)
and the size of the population \(P\) given by
\begin{equation}D = d \cdot P. \label{men-5}\tag{7.1.2}\end{equation}
These models are often characterized by saying the population has a
*constant per capita rate*.

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Example7.1.11

The United States Census Bureau's summary of births reports a birth rate
of 14.0 births per thousand individuals in the population for the year 2008.
The population that year was 304.1 million individuals. How many births
were recorded in 2008?

This vital statistic information allows us to compute the per capita birth rate
based on a population of 1000.
\begin{equation*} b = \frac{B}{P} = \frac{14.0}{1000} = 0.0140 \end{equation*}
For the full population \(304.1\times 10^{6}\), we can compute the
birth rate for the year.
\begin{equation*} B = b \cdot P = 0.0140 (304.1 \times 10^{6}) = 4.26 \times 10^{6} \end{equation*}
There were about 4.26 million births during the year 2008.

Checking the bureau's direct report, we can find there were 4.248 million births
during the year. The discrepancy comes from reporting the per capita birth rate
with only three significant digits. A more accurate value would have been \(b=13.97\)
which leads to a matching result.

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Subsubsection7.1.3.2Cost and Unit Cost

Many goods and services are purchased with a fixed unit cost.
This is another way of saying that the total cost is proportional
to the measure of the goods or services purchased.
Someone working at a fixed hourly rate receives a payment that is proportional
to the number of hours worked.
Grocery stores give prices for produce in dollars per pound, meaning that
the cost of purchase is proportional to the weight.

When multiple items are purchased, the total cost is the sum of the individual
costs. In this way, if there are different unit costs, we can still compute
the total cost.

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Example7.1.12Material Costs

You are finishing a room that is 10 feet by 18 feet and a ceiling that
is 8 feet high. There are two windows (3 feet by 5 feet) and one door
(3 feet by 7 feet). If wall paper costs $0.60 per square foot
and carpet costs $1.25 per square foot, what will be the total
cost of materials (ignoring unused scraps)?

Notice that cost information is given per square foot. These prices are
the proportionality constants between the cost and the area. So we first need
to find the areas and then compute the cost.

The carpet is a rectangle that is 10 feet by 18 feet. The area is computed
as length times width,
\begin{equation*}A_{\hbox{carpet}} = L \cdot W = 10(18) \; \hbox{ft}^2 = 180 \; \hbox{ft}^2,\end{equation*}
and the carpet cost is proportional,
\begin{equation*}C_{\hbox{carpet}} = 1.25 \; \hbox{dollars/ft}^2 \cdot A_{\hbox{carpet}}
= 225 \; \hbox{dollars}.\end{equation*}
Notice how the unit price divides by a dimension of area which when multiplied
by the area (in the same units) cancel. If the price was given in different
units, we would first need to perform unit conversion.

The walls consist of four rectangles which are then missing three smaller
rectangles (for the door and windows).
Two walls have width 10 feet and height 8 feet for area \(80 \; \hbox{ft}^2\) each;
the other walls have length 18 ft and height 8 ft for area \(144 \; \hbox{ft}^2\) each.
The door removes \(21 \; \hbox{ft}^2\) and each window removes
\(15 \; \hbox{ft}^2\). The total area needing wallpaper is
\begin{equation*}A_{\hbox{wallpaper}} = [2(80) + 2(144) - 21 - 2(15)] \; \hbox{ft}^2 = 397 \; \hbox{ft}^2,\end{equation*}
with a resulting proportional cost
\begin{equation*}C_{\hbox{wallpaper}} = 0.60 \; \hbox{dollars/ft}^2 \cdot A_{\hbox{wallpaper}}
= 238.20 \; \hbox{dollars}.\end{equation*}
The total cost is the sum
\begin{equation*}C = C_{\hbox{carpet}} + C_{\hbox{wallpaper}} = 463.20 \; \hbox{dollars}.\end{equation*}