Applications of differential equations:
1. A 2000 gallon tank can support no more than 150 guppies. Six guppies are introduced into the
tank. Assume that the rate of growth of the population is
dP
= 0.0015P(150 - P),
dt
where t is in weeks.
a. Write a function of t for guppy population.
b. How long will it take for the guppy population to reach 100?
2. A certain wild animal preserve can support no more than 250 lowland gorillas. Twenty-eight
gorillas were known to be in the preserve in 1970. Assume that the rate of growth of the
population is
dP
= 0.0004 P(250 - P),
dt
where t is in years.
a. Write a function of t for gorilla population.
b. How many more years will it take for the gorilla population to reach the carrying
capacity?
dp
at which p changes
dh
with the altitude h above sea level is proportional to p. Suppose that pressure at sea level is 1013
millibars (1 atmosphere) and that the pressure at an altitude of 20 km is 90 millibars.
a. Solve the initial value problem:
dp
Differential equation:
= kp ,
dh
Initial condition: p = p0 when h = 0 ,
to express p in terms of h. Determine constants from the altitude-pressure information provided.
b. What is the atmospheric pressure at h = 50 km?
c. At what altitude does the pressure equal 900 millibars?
3. Earth’s atmospheric pressure p is often modeled by assuming the rate
4. A tank contains 1000L brine with 15 kg of dissolved salt. Pure water enters the tank at a rate
of 10L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate.
a. Write an equation for the amount of salt as a function of time.
b. How much salt is in the tank after 20 minutes?
5. A tank contains 1000L of pure water. Brine with 0.05 kg/L salt enters the tank at 5L/min.
Brine with 0.04 kg/L salt enters at a rate of 10 L/min. The solution is kept thoroughly mixed and
drains from the tank at a rate of 15 L/min.
a. Write an equation for the amount of salt as a function of time?
b. How much salt is in the tank after one hour?
6. A hard-boiled egg at 98 °C is placed in a bowl with running cold water (18 °C) so that it will
cool. After 5 minutes, the temperature of the egg is 38 °C. How much longer will it take for the
egg to cool to 20 °C?
7. Suppose that electricity is draining from a capacitor at a rate proportional to the voltage V
across its terminals, with t measured in seconds,
dV
1
=- V
dt
40
a. Solve this differential equation for V, using V0 to denote the value of V when t = 0 .
b. How long will it take the voltage to drop to 10% of its original value?
8. A radioactive material will decay at a rate proportional to the amount of the material. Charcoal
from a tree killed in the volcanic eruption that that formed Crater Lake in Oregon contained
44.5% of the C14 found in living matter. About how old is Crater Lake? The half-life of C14 is
5700 years.