19-07-2012, 04:31 PM
Applications of Differential Equations
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Modeling Advertising Awareness
The new cereal product from Example 3 in Section C.1 is introduced through an
advertising campaign to a population of 1 million potential customers. The rate at
which the population hears about the product is assumed to be proportional to the
number of people who are not yet aware of the product. By the end of 1 year, half
of the population has heard of the product. How many will have heard of it by the
end of 2 years?
Solution Let y be the number (in millions) of people at time t who have heard of
the product. This means that is the number of people who have not heard,
and is the rate at which the population hears about the product. From the
given assumption, you can write the differential equation as follows.
Modeling a Chemical Reaction
During a chemical reaction, substance A is converted into substance B at a rate
that is proportional to the square of the amount of A. When 60 grams of A
are present, and after 1 hour only 10 grams of A remain unconverted.
How much of A is present after 2 hours?
Solution Let y be the unconverted amount of substance A at any time t. From the
given assumption about the conversion rate, you can write the differential equation
as follows.
Modeling Population Growth
A population of 20 wolves has been introduced into a national park. The forest
service estimates that the maximum population the park can sustain is 200
wolves. After 3 years, the population is estimated to be 40 wolves. If the population
follows a Gompertz growth model, how many wolves will there be 10 years
after their introduction?
Modeling Hybrid Selection
You are studying a population of beetles to determine how quickly characteristic
D will pass from one generation to the next. At the beginning of your study
you find that half the population has characteristic D. After four generations
you find that 80% of the population has characteristic D. Use the
hybrid selection model above with and to find the percent of the
population that will have characteristic D in 10 generations.