1. (25 points) A 180-lb skydiver drops from a hot-air balloon. After 10 sec of free fall, a parachute is opened. The parachute immediately introduces a drag force proportional to the velocity. After an additional 4 sec, the parachutist reaches the ground. Assume the air resistance is negligible during free fall and that the parachute is designed so that a 200lb person will reach a terminal velocity of 10 mph . ( g = 32 f t/sec 2 , 1 mile = 5280 f eet )
(a) What is the speed of the skydiver immediately before the parachute is opened?
(b) What is the parachutist impact velocity?
(c) At what altitude was the parachute opened?
(d) What is the balloon altitude?

2. (25 points) Consider a bioreactor used by a yogurt factory to grow the bacteria needed to make yogurt. The growth of the bacteria is governed by the logistic equation dP dt
= k · P ( M − P ) where P is the population in millions and t is the time in days. Recall that M is the carrying capacity of the reactor and k is a constant that depends on the growth rate.
(a) Through observation it is found that after a long time the population in the reactor stabilizes at 30 million bacteria, and that when the population of the reactor is 10 million the population increases at a rate of 2.1 million bacteria per day. From this, find k and M in the governing equation.
(b) If the colony starts with a population of 500,000 bacteria, how long will it take for the population to reach 85% of the carrying capacity?
(c) Suppose the factory harvests bacteria from the reactor once a week. The harvesting process takes a day, during which the reactor is not operational, leaving 6 days per week for the bacteria to grow in the reactor. The factory wants to maximize the amount of bacteria grown during these 6 days. To achieve this, dP should be at its dt maximum 3 days after harvesting. What initial population (after harvesting) gives the most growth over the 6-day period? What is the population change during that time?
(d) Suppose the reactor is modified to allow for continual harvesting without shutting down the reactor. Let h be the rate at which the bacteria are harvested in millions per day. Write down the new differential equation governing the bacteria population. What is the maximum rate of harvesting h that will not cause the population of bacteria to go extinct? (Below this rate there will always be a stable equilibrium point where P is positive).
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3. (25 points) Suppose that a given population can be divided into two parts: those who have a given disease and can infect others and those who do not have it but are susceptible. Let x be the proportion of susceptible individuals and y the proportion of the infectious individuals; then x + y = 1. Assume that the disease spreads by contact between sick and well members of such contacts. Further, assume that members of both groups move about freely among each other, so that the number of contacts is proportional to the product of x and y . Since x = 1 − y , we obtain the differential equation: dy
= αy (1 − y ) , y (0) = y
0 dt where α is a positive number.
(a) Find the equilibrium points for the given differential equation and draw the phase diagram to determine whether each is stable or unstable.
(b) Solve the initial value problem given y (0) = y
0 and verify the conclusion you reached in part (a) is correct. Show that y ( t ) → 1 as t → ∞ .
Hint: you’ll need to use partial fractions.
(c) Suppose a constant h proportion of the population is recovering themselves constantly and α = 1, i.e. the differential equation becomes dy
= y (1 − y ) − h, y (0) = y
0
.
dt
For what values of h and y
0 will this population all get infected? or all get recovered?
Draw the diagram respectively.
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