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(Due date: 6:00pm, October 4, 2013 (Friday))
1. Digoxin is used in the treatment of heat disease. In the following table, y represents
the amount of digoxin in the bloodstream and t represents the time in days after taking asingle dose. The initial dosage is 0.5 mg.
(i) Formulate a discrete model for the change in concentration per day being proportional
to the amount of digoxin present. Test your assumption by plotting the change versus theamount present at the beginning of the period.
(ii) Assume that after the initial dose of 0.5 mg, each day a maintenance dose of 0.1 mg
is taken. Formulate the refined model.
(iii) Build a table of values or numerical solution for (ii) for 15 days.
2. Consider a reservoir with a volume of 8 billion cubic feet (ft3) and an initial pollutant
concentration of 0.25%. There is a daily inflow of 500 million ft3 of water with a pollutantconcentration of 0.05% and an equal daily outflow of the well-mixed water in the reservoir.
How long will it take to reduce the pollutant concentration in the reservoir to 0.10%?
3. Sociologists recognize a phenomenon called social diffusion, which is the spreading of a
piece of information, a technological innovation, or a cultural fad among a population. Themembers of the population can be divided into two classes: those who have the informationand those who do not. In a fixed population whose size is known, it is reasonable to assumethat the rate of diffusion is proportional to the number who have the information times thenumber yet to receive it. If X(t) denotes the number of individuals who have the informationat time t in a population of N people, then a mathematical model for social diffusion is givenby
(i) Solve the model and show that it leads to a logistic curve.
(ii) At what time is the information spreading fastest?(iii) How many people will eventually receive the information?(iv) Find the equilibrium and their stability.
4. Consider a rabbit population N(t) satisfying the logistic equation. If the initial
population is 120 rabbits and there are 8 births per month and 6 deaths per month occurringat time t = 0, how many months does it take for N(t) to reach 95% of the limiting populationK? (Hint: for the logistic model dN(t) = a N
− b N 2, B = a N is the time rate at which
births occur and D = b N2 is the time rate at which deaths occur. If the initial populationN(0) = N0, and B0 births per time and D0 deaths per time are occurred at time t = 0, thenthe limiting population is K = B0N0/D0).
5. The terminal speed of a bomb is found to be 400 km/h by experiment. The designers
wish the bomb to explode at an altitude of 600 meters after being dropped from 10,000 metersin horizontal flight. At how many seconds should the time delay of the firing mechanism beset if the mechanism is activated when the bomb leaves the plane? To simply the model,ignore the horizontal movement and assume that the air resistance is proportional to thesquare of the speed.
6. A rocket is launched in a vertical direction. The total mass is 25,000 kg of which 20,000
kg is fuel. The engines emit exhaust gases at a constant rate of 40,000 meters per secondand consume fuel at a constant rate of 100 kg per second. The resistance is −200v Newtonwith v(t) the velocity of the rocket (measured in meters per second) at time t. Assume thatthe acceleration of gravity is 9.8 m/s2 during the flight.
(i) Show that the following initial value problem is valid until all the fuel is consumed:
(ii) Solve the above initial value problem.
(iii) Calculate v at the moment when all the fuel had been consumed. What do you infer
from your answer as regards the motion of the rocket?
7. A patient is given a dose Q of a drug at regular intervals of time T . The concentration
of the drug in the blood has been shown experimentally to obey the law
(i) If the first dose is administered at t = 0 hr, show that after T hr have elapsed, the
(ii) Assuming an instantaneous rise in concentration whenever the drug is administered,
show that after the second dose and T hr have elapsed again, the residual
(iii) Show that the limiting value R of the residual concentrations for doses of Q mg/ml
repeated at intervals of T hr is given by the formula
(iv) Assuming the drug is ineffective below a concentration L and harmful above some
higher concentration H, show that the dose schedule T for a safe and effective concentrationof the drug in the blood satisfies the formula
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