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Variation and Mathematical Modeling
When we know the relationship between two or more variables, we can then write an
equation relating the variables to each other.
In the sciences and business there are
many formulas that describe the variation between quantities relating to the application.
We refer to these formulas as
mathematical models
of the application.
In order to
formulate the model, we need to understand
what type of variation
is occurring.
Direct Variation
When two variables, x & y vary directly, then
y = k
•
x
where
k is a constant
called the
"constant of proportionality." When there is direct variation between variables x and y,
doubling the value of x will always result in y being doubled.
Also, if x is cut in half, then
y is cut in half.
An Example of Direct Variation
If
y represents the miles traveled by a person driving down the highway at a speed of 50
MPH and
x represents the hours spent traveling, then y = 50x and x and y vary directly.
We also say that x is directly proportional to y.
So if the person travels for x = 2 hours,
they will cover y =
50(2) = 100 miles.
If
the person travels for x = 4 hours, then they will
cover y = 50(4) = 200 miles. A doubling of the hours x results in a doubling of the miles y.
Example: In chemistry, we learn that for an ideal gas, pressure P varies directly as
temperature T if all other variables are held constant. What is the equation relating
P to T?
Here, P is related to T in the same way x is related to y in the definition of direct
variation. So we can write:
P = k
•
T
.
Another acceptable answer would be
T = k
•
P
.
Example: If S varies directly as T, and S = 40 when T = 5, write an equation relating
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 Fall '09
 Crandall
 Anthropology, Mass, Mathematical constant

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