MAP 2302 In-Class Activity #1 - Fall 2011
(1) Let P (t) represent the density of a population at time t. Suppose the population
grows with an intrinsic growth rate of r, has an environmental carrying capacity
of K, and an initial population density of P (0) = P0 . Then the rate at which the
population’s density changes with time can be modeled by the logistic equation
P
dP
= rP 1 −
.
dt
K
(a) If P > K (i.e., the population density if greater than the environmental
carrying capacity), what does the model say will happen to the population’s
density? If P < K (i.e., the population density is less than the environmental
carrying capacity, what does the model say will happen to the population’s
density?
(b) Determine what will happen to the population density in the limit as P → K.
Explain why this makes sense ecologically.
Before we solve the logistic equation, let’s make the following substitutions:
P = Ky
and
1
t= x
r
(1)
Using the chain rule, we get
dP dy dx
dP
=
·
·
dt
dy dx dt
dy
=K·
·r
dx
Substituting this into the logistic equation, along with y and x from (1) gives
Ky
dy
Kr
= r(Ky) 1 −
.
dx
K
After simplifying we are left with
dy
= y(1 − y)
dx
(2)
This equation is referred to as the non-dimensionalized logistic equation and there
are multiple ways to solve it.
(c) View equation (2) as a Bernoulli equation and solve it for y(x). Then use the
substitutions in (1) to get P (t). Note: You need to switch y and x back to P
and t before you use the initial condition.
(d) Determine the eventual fate of the population; that is, determine
lim P (t).
t→∞
(2) Carbon dating in a technique used to determine the age of an object or artifact.
The idea is based on the radioactive decay of carbon-14 atoms. What happens
is that cosmic radiation (such as from the sun) enters the Earth’s atmosphere
causing a chemical reaction that leads to carbon-14. It is well-known that all
living organisms contain both carbon-12 (which is necessary for life) and carbon14 (which is absorbed from the atmosphere). Scientists believe that the amount
of carbon-14 in the atmosphere is constant, so the ratio of carbon-12 to carbon14 in all living organisms remains constant constant while the organism is alive
(while carbon-14 decays, the living organism is able to absorb more by means
of eating to replace what has been lost). When the organism dies, the amount
of carbon-12 remains the same, but carbon-14 decays without being replenished.
By comparing the ratio of carbon-12 to carbon-14 in a dead organism to that of
a living organism, scientists can determine how much carbon-14 has decayed and
hence how long the organism has been dead for.
(a) Let A(t) represent the amount of carbon-14 which remains in an organism
after t years. Suppose the initial amount of carbon-14 is A(0) = A0 and
that it has a decay constant of k. The rate at which carbon-14 decays is
proportional to the amount remaining, so radioactive decay of carbon-14 can
be modeled by the equation
dA
= kA.
dt
Solve this equation as a first-order linear equation to determine A(t).
(b) The half-life of an atom is the amount of time it takes for the atom to decay to
half of its original amount. In other words, the half-life is the time it takes for
A(t) = 21 A0 . It is known that the half-life of carbon-14 is approximately 5700
years. Use this information to determine the decay constant for carbon-14.
(c) Suppose an archaeologist finds a bone believed to belong to an animal which
lived approximately 24,000 years ago. If this archaeologist is correct, then
what percent of the original carbon-14 should be found in the bone when he
takes it back for analysis?
(d) Suppose the FBI is investigating a 1000 year old painting which was stolen
from a museum. They receive a tip that someone is selling the painting
online. When they arrive at the seller’s house, he claims that the painting is
a duplicate made in the 1900’s. When the FBI tests the wood in the frame,
they discover that 88.5% of the carbon-14 in the wood is still present. Should
the FBI believe the seller’s story or should they suspect that the painting
might be stolen?
(3) According to Newton’s Second Law of motion,
F = ma.
We know that velocity and acceleration are related by the equation a = dv
, so we
dt
can rewrite Newton’s Second Law of motion as
dv
F =m .
dt
Suppose a skydiver jumps out of an airplane with initial vertical velocity of v(0) =
0. Then the two forces acting on him are the force of gravity, Fg , and air resistance,
FR . Since we are concerned with a falling body, we can take the positive direction
to be downward, so the force due to gravity is
Fg = mg.
The force due to air resistance is in the opposite direction of motion, so it will
be negative. Empirical evidence suggests that at low speeds, air resistance is
proportional to velocity and for high speeds, air resistance is proportional to
velocity squared. In the case of a skydiver, this means that the force due to air
resistance before opening the parachute is
FR = −kv 2
and after opening the parachute is
FR = −mv.
Thus, a model for the velocity of a skydiver before opening his parachute is
dv
(3)
m = mg − kv 2
dt
and for a skydiver after opening his parachute is
dv
(4)
m = mg − kv.
dt
k
If we divide both sides of both equation by m and make the replacement D = m
(D is referred to as the drag coefficient) then equations (3) and (4) become
dv
dv
= g − Dv 2
and
= g − Dv.
dt
dt
(a) Solve both equations (you don’t have to use the same method for both) to
get v(t).
(b) Determine the terminal velocity of the skydiver in both cases; that is, determine
lim v(t).
t→∞
(c) Suppose you have been contracted to design parachutes for the U.S. Army’s
82nd Airborne Division. They want you to design a parachute that will enable
their soldiers to land at under 15 mph (22 ft/sec). What is the minimal drag
coefficient you can build your parachutes to have?
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