APM1514 Assignment 4 (COMPLETE ANSWERS) 2024 - DUE 10 June 2024

APM1514 Assignment 4 (COMPLETE ANSWERS) 2024 - DUE 10 June 2024 ;100 % TRUSTED workings, explanations and solutions. For assistance call or W.h.a.t.s.a.p.p us on ...(.+.2.5.4.7.7.9.5.4.0.1.3.2)........... Question 1: 8 Marks Suppose that the population of South Africa was approximately 60, 04 million at the beginning of year 2021, and that it follows the Malthusian model approximately, with a growth rate of 1.24 percent per year. (1.1) When will the population reach 70 million, if the growth rate does not change? (4) (1.2) What does the model predict the population of South Africa to be in the year 2050? (4) Question 2: 14 Marks (2.1) Given H = 2 mg/ml, L = 0.5 mg/ml, and k = 0.02 h (7) − 1 , suppose concentrations below L are not only ineffective but also harmful. Determine a scheme for administering this drug (in terms of concentration and times of dosage). (2.2) Suppose that k = 0.3hr (7) − 1 and that the smallest effective concentration is 0.04 mg/ml. A single dose that produces a concentration of 0.1 mg/ml is administered. Approximately how many hours will the drug effective? Question 3: 10 Marks The populations of Country R and Country U both grow according to the Malthusian model, Country R with a doubling time 100 years and Country U with a doubling time 120 years. If the sizes of the populations were the same in year 2000, what was the ratio of the population of Country U to the population of Country R in 1900? What will the ratio be in 2100? Question 4: 12 Marks Which of the following statements are true, and which are false? You must be able to justify your answers! (4.1) The more there is of a radioactive substance, the greater its rate of decay is. (4) (4.2) If the growth constants of two Malthusian populations A and B are kA and kB, and if kA = (4) 2 ∗kB, then after 10 years the population A is two times larger than the population B. 30 APM1514/101/0/2024 (4.3) If the growth constants of two Malthusian populations A and B are kA and kB, and if kA = (4) 2 ∗kB, then the population A takes twice as long to double as population B. Question 5: 10 Marks Measurements show that 200 kg of radioactive waste removed from a nuclear reactor in 1970 has decayed to 195 kg in 1995. (5.1) How much of this radioactive material will be left in the year 2095? (5) (5.2) When will there be only 20 kg left? (5) Question 6: 12 Marks In each of the following cases of parameter values, draw the phase line of the corresponding logistic model and read from the phase line the outcomes for a solution which starts at (i) P0 = 100, (ii) P0 = 1000. (6.1) a = 0.001, b = 0.0005, (6.2) a = 0.1, b = 0.0003, (6.3) a = 0.01, b = 5× 10− 6 . Question 7: 18 Marks Assume that P(t )follows logistic growth, with initial valueP0 = 200, a = 0.5 and b = 0.02. (7.1) Find P(1) and P(2). (4) (7.2) Find the rate of change of P at the times t = 0, t = 1 and t = 2. (6) (7.3) Draw the phase line of this population model. (4) (7.4) Plot a sketch of the solution curve P(t ) when P0 = 200, over the interval 0 ≤ t ≤ 3. Use (4) the information in your answers to (7.1), (7.2) and (7.3) to make the sketch as accurate as possible. Question 8: 16 Marks (8.1) The population of a country, denoted by P(t ) with t measured in years, grows according to the logistic model with a = 0.3 and with an initial population size of 6 × 106 . Calculate the value of b, in the following two cases: (a) If the annual increase from t = 0 to t = 1 is 10%. (8) (b) If the rate of change of the population at time t = 0 is 10% of the population per unit (8) time