Applications of Differential Equations:
Exponential and Logistic
Chapter: 7 Assign: 26
In exponential growth, the rate of change of the quantity is proportional to the quantity itself. In logistic
growth, this is also true, BUT the quantity is also proportional to the distance from the upper bound.
This translates to the formula
dy
= ky( L − y) , where L is the carrying capacity (upper bound) and k is
dt
the constant of proportionality. If you solve this differential equation using partial fraction
decomposition, you get the general solution€y =
€
L
1 + ce−Lkt
. The actual solving is a BC topic, but in
our class you are expected to know and be comfortable working with both formula’s above.
1. The Easter Bunny has begun to express his malevolent side. This year, instead of hiding real eggs,
€ substance, Nb-95, which has a half-life of 35 days. If the danger
he’s hiding eggs made of a radioactive
eggs have a mass of 2 kg and you don’t find one hiding under your bed, how long will it take one of
these eggs to decay to harmless, 50gms?
2. A radioactive isotope has a half-life of 16 days. You wish to have 30g at the end of the 30 days. How
many grams should you start with?
3. A highly contagious “pinkeye”, Conjunctivitus itchlikecrazius, is ravaging the school. The population
of the school is 1440 including students and staff and the rate of infection is proportional to BOTH the
number of students infected AND the number of students not yet infected. If 75 people were infected on
December 15th and 250 have contracted it by December 20th, how many people will celebrate New
Year’s having been infected?
4. U-235 is an unstable isotope of U-238. Both are found in the southwest and both are used in nuclear
weapons. The half-life of U-235 is 710 million years and the warhead in the Minuteman W62 warhead
is 253 pounds. Create a model that tells you how many pounds of U-235 you will have after t years.
5. Sarah likes mollies above all other tropical fish. Her fish must really like each other because they are
reproducing like crazy. The rate of increase of the fish population is proportional to both the current
population of mollies and the number of additional fish the tank can support. Her tank has a carrying
capacity of 50 mollies. If she bought 10 fish to start the tank two months ago (none of which died) and
she has 25 fish now, how many fish will she have in one month?
6. For what value of y is the rate of change greatest for logistic model? Use the models on at the
beginning of this sheet to help justify your answer. You can’t just give an answer, but must show why
that is the answer.
7. A human zygote consists of one cell at conception and the number of cells grow to 8 by the end of
one week. Assuming that the rate of cell increase is proportional to the number of cells, how many
weeks will it take for there to be 1000 cells?
8. Newton’s Law of Cooling states that an object cools down proportional to the difference between its
temperature and the temperature of the surroundings. If my cocoa was 100˚F 10 minutes ago is now
only 90˚F, what will the temperature of the cocoa be in 15 min? The temperature of the room is 75˚F.
9. The rate of change in population of Irukandji jellyfish off the coast of Australia is directly
proportional to 450000 − P , where P is the current population of Irukandji and t is time in years. If
there were 100,000 in one colony off Cairns, Australia in 2000 and there were 180,000 two years later
find a. what the population of jellyfish in this colony was in 2006 and b. In what year the population
€
would reach 300,000 , considered
enough to take over the local area from other aquatic species.
€
€ builds 5 homes and then opens
10. A real estate developer opens up a subdivision with 120 plots. She
up the lots to other builders. The number of homes built on starts slowly, increases faster and then
€ peters out as the last lots are built on. The rate of change in the number of homes is proportional
slowly
to the number of homes as well as the number of lots remaining. If the constant of proportionality is
.0075, when will 70% of the plots have homes on them? What is the rate of change in the number of
homes at this moment? Time is in months.
11. The Bighorn sheep in the mountains around Tucson have a rate of change in population with respect
to time is proportional to 100 − P . Due to human population expansion and the concomitant release of
large numbers of domestic goats to graze in the area, the number of Bighorn sheep in the 1990s was
down to 70. (In the year 2000, sheep population is 70) A plan to increase the population by studying
€ and helping to carve out transit routes for them between the three Tucson mountain
their interactions
ranges. In 2004, despite an increase in the spread of conjunctivitis from domestic goats which killed a
number of them (ha! And you thought pinkeye was just a joking matter) the population was at 74.
When will the population reach 80?