Bacteria Problem: Harry and Hermione start a culture of bacteria in a laboratory dish by

Chapter 7, Problem 1

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Bacteria Problem: Harry and Hermione start a culture of bacteria in a laboratory dish by introducing 3 million bacteria at time t = 0 hours. The number of bacteria increases rapidly at first, then levels off. The two lab partners estimate that the dish can support a maximum of 30 million bacteria. They let B stand for the size of the bacteria population, in millions, at time t, in hours, and assume that the rate of change of B is given by the logistic differential equation The slope field for this differential equation is shown in Figure 7-6c. Figure 7-6c a. Explain the real-world influence on the rate of bacteria growth of the two factors B and (30 B)/30 in the differential equation. Explain why it is reasonable that dB/dt is positive for 0 B 30 and negative for B 30 b. On a copy of Figure 7-6c, sketch the graphof the particular solution subject to thegiven initial condition of 3 million bacteria.Also, sketch the graph of the particularsolution if Harry and Hermione try to speedup the process by adding enough bacteria attime t = 10 to make a total of 40 millionbacteria. What is the major difference in thebehavior of the population for the twodifferent initial conditions?c. Use Eulers method with dt = 0.5 and initialcondition (0, 3) to estimate B at times t = 10,20, 30, and 40 h. Mark these points on yourgraph from part b. Does your graph agreereasonably well with these values?d. Show the steps you used to solve thedifferential equation algebraically, subject tothe initial condition that B = 3 when t = 0.Write B explicitly as a function of t. Use theparticular solution to find out how close theEulers method estimate of B is to theprecise predicted value of B when t = 20.e. Use the differential equation to show thatthe greatest rate of increase in the bacteriapopulation occurs when B is halfwaybetween the horizontal asymptotes at B = 0and B = 30. Find the value of t at this pointof inflection.