Hello I need help with these calc problems

1400 EXAMPLE 1 Draw a direction field for the logistic equation with k = 0.09 and carrying capacity M = 1000. What can you deduce about the solutions? 1200 1000 SOLUTION In this case the logistic differential equation is dp 800 dt = 0.09P 1 - P(t) 1000 600 A direction field for this equation is shown in the top figure. We show only the first quadrant because negative 400 populations aren't meaningful and we are interested only in what happens after t = 0. 200 The logistic equation is autonomous (dP/dt depends only on P, not t), so the slopes are the same along any vertical line. As expected, the slopes are positive for 0 20 40 60 80 The slopes are small when P is close to 0 or (the carrying capacity). Notice that the solutions move away from direction field for the logistic equation the equilibrium solution P = 0 and move toward the equilibrium solution P = 1400 In the bottom figure we use the direction field to sketch solution curves with initial populations P(0) = 100, P(0) = 400, 1200 and P(0) = 1200. Notice that solution curves that start below P = 1000 are ( ---Select--- @ and those that start above P = 1000 are ---Select--- @ . The slopes are greatest when P 2 500. In fact, we can prove that all solution curves that 1000- start below P = 500 have an inflection point where P is exactly 500. 800 P(t) 600 400 200 20 40 60 80 solution curves for the logistic equation8. [-/1 Points] DETAILS SCALCET8 9.4.010. MY NOTES PRACTICE ANOTHER (a) Assume that the carrying capacity for the US population is 800 million. Use it and the fact that the population was 282 million in 2000 to formulate a logistic model for the US population. (Let t = 0 correspond to the year 2000. Use k for your constant.) P(t) = millions (b) Determine the value of k in your model by using the fact that the population in 2010 was 309 million. K = (c) Use your model to predict the US population in the years 2300 and 2400. (Round your answers to the nearest integer.) year 2300 million year 2400 million (d) Use your model to predict the year in which the US population will exceed 700 million. (Round your answer down to the nearest year.) Need Help? Read It Submit