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7.5. MODELING WITH DIFFERENTIAL EQUATIONS
Developing a differential equation
Preview activity 7.5 demonstrates the kind of thinking we will be doing in this section. In
each of the two examples we considered, there is a quantity, such as the amount of money
in the bank account or the amount of salt in the tank, that is changing due to several
factors. The governing differential equation results from the total rate of change being the
difference between the rate of increase and the rate of decrease.
Example 7.3. In the Great Lakes region, rivers flowing into the lakes carry a great deal of
pollution in the form of small pieces of plastic averaging 1 millimeter in diameter. In order
to understand how the amount of plastic in Lake Michigan is changing, construct a model
for how this type pollution has built up in the lake.
Solution.
First, some basic facts about Lake Michigan.
• The volume of the lake is 5 · 10 12 cubic meters.
• Water flows into the lake at a rate of 5 · 10 10 cubic meters per year. It flows out of
the lake at the same rate.
• Each cubic meter flowing into the lake contains roughly 3 · 10 −8 cubic meters of
plastic pollution.
Let’s denote the amount of pollution in the lake by P(t), where P is measured in cubic
meters of plastic and t in years. Our goal is to describe the rate of change of this function;
in other words, we want to develop a differential equation describing P(t).
First, we will measure how P(t) increases due to pollution flowing into the lake. We
know that 5 · 10 10 cubic meters of water enters the lake every year and each cubic meter
of water contains 3 · 10 −8 cubic meters of pollution. Therefore, pollution enters the lake at
the rate of
5 · 10
10 m 3 water
year
·
3 · 10
−8 m 3 plastic
m 3 water
= 1.5 · 10
3 cubic m of plastic per year.
Second, we will measure how P(t) decreases due to pollution flowing out of the lake.
If the total amount of pollution is P cubic meters and the volume of Lake Michigan is
5 · 10 12 cubic meters, then the concentration of plastic pollution in Lake Michigan is
P
5 · 10 12 cubic meters of plastic per cubic meter of water.
7.5. MODELING WITH DIFFERENTIAL EQUATIONS
Developing a differential equation
Preview activity 7.5 demonstrates the kind of thinking we will be doing in this section. In
each of the two examples we considered, there is a quantity, such as the amount of money
in the bank account or the amount of salt in the tank, that is changing due to several
factors. The governing differential equation results from the total rate of change being the
difference between the rate of increase and the rate of decrease.
Example 7.3. In the Great Lakes region, rivers flowing into the lakes carry a great deal of
pollution in the form of small pieces of plastic averaging 1 millimeter in diameter. In order
to understand how the amount of plastic in Lake Michigan is changing, construct a model
for how this type pollution has built up in the lake.
Solution.
First, some basic facts about Lake Michigan.
• The volume of the lake is 5 · 10 12 cubic meters.
• Water flows into the lake at a rate of 5 · 10 10 cubic meters per year. It flows out of
the lake at the same rate.
• Each cubic meter flowing into the lake contains roughly 3 · 10 −8 cubic meters of
plastic pollution.
Let’s denote the amount of pollution in the lake by P(t), where P is measured in cubic
meters of plastic and t in years. Our goal is to describe the rate of change of this function;
in other words, we want to develop a differential equation describing P(t).
First, we will measure how P(t) increases due to pollution flowing into the lake. We
know that 5 · 10 10 cubic meters of water enters the lake every year and each cubic meter
of water contains 3 · 10 −8 cubic meters of pollution. Therefore, pollution enters the lake at
the rate of
5 · 10
10 m 3 water
year
·
3 · 10
−8 m 3 plastic
m 3 water
= 1.5 · 10
3 cubic m of plastic per year.
Second, we will measure how P(t) decreases due to pollution flowing out of the lake.
If the total amount of pollution is P cubic meters and the volume of Lake Michigan is
5 · 10 12 cubic meters, then the concentration of plastic pollution in Lake Michigan is
P
5 · 10 12 cubic meters of plastic per cubic meter of water.