The Pacific halibut fishery has been modeled by the differential equation dt dy = ky(1 - where y(t) is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be M = 7 x 10 kg, and k = 0.79 per year. (a) If y(0) = 2 x 10' kg, find the biomass a year later. (Round your answer to two decimal places.) 3.44 X x 10 kg (b) How long will it take for the biomass to reach 4 x 10 kg? (Round your answer to two decimal places.) 1.30 X yr Need Help? Read It Watch It Master ItThe population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 120 billion. (Assume that the difference in birth and death rates is 20 million/year = 0.02 billion/year.) (a) Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate. Let P be the population in billions and t be the time in years, where t = 0 corresponds to 1990.) _ LP(l L) dt 265 120 v (b) Use the logistic model to estimate the world population in the year 2000. Compare with the actual population of 6.1 billion. (Round the answer to two decimal places.) P= % billion (c) Use the logistic model to predict the world population in the years 2100 and 2500. (Round your answer to two decimal places.) year 2100 P = X billion year 2500 P = billion (d) What are your predictions if the carrying capacity is 60 billion? (Round your answers to two decimal places.) year 2000 P = billion year 2100 P = X billion year 2500 P = billion Need Help? [omsan] [ vmens ] e ] One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. (a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.) gy _ dt (b) Solve the differential equation. (Let y(0) = yq.) (c) A small town has 4000 inhabitants. At 8 AM, 320 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Do not round k in your calculation. Round your final answer to one decimal place.) hours after 8 AM Need Help? [ZResaie ] [Cwatenie ) Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 9 sin(t), y = 8 cos(t), -Tt S t s 7n This answer has not been graded yet.(a) Show that the parametric equations x = X1+ (X2 - X1)t, y = y1+ ()2 - y1)t where 0 S t s 1, describe the line segment that joins the points P1(X1, y1) and P2(X2, Y2). This answer has not been graded yet. (b) Find parametric equations to represent the line segment from (-3, 5) to (2, -3). (Enter your answer as a comma-separated list of equations. Let x and y be in terms of t.)