1. [/8 Points] DETAILS MY NOT ' (a) Consider the following logistic growth equation. \" : 3A" (3 I\" Determine the carrying capacity. (Give an exact answer) Determine intrinsic growth rate. (Give an exact answer.) (b) Consider the following logistic growth equation. ri' _ _ phi0.0 2 1\" Determine the carrying capacity. (Give an exact answer.) Determine intrinsic growth rate. (Give an exact answer.) Submit Answer 2. [I6 Points] DETAILS MY NOTE! The growth of a certain type of weed satisfies a differential equation of the following form, where a is the height of the weed at maturity: 52(dY)/(dt) : Marv) Thus we know that the height, y(t), of the weed after t days have passed is given by the following equation: a r:7 y() 1+Be'5k' If a typical Weed starts at height 9 centimeters, has height 50 centimeters after 22 days, and reaches height 55 centimeters at maturity, find the Values of the constants B and k (correct to at least 4 decimz places). Suhmlt Answer 3. [-I18 Points] MY NOT A news item is spreading by word of mouth through a population of size 30,000 people, After rdays, the number of people (in thousands) who have heard the news is given by the following equation. W:f(t):(30)/(1+150 e\"(cl,2 t)) (a) Approximately how many thousand people have heard the news after 8 days? (Give your answer correct to at least three decimal places.) thousand people (b) Calculate the derivative f'(r), m) : (c) At what rate is the news spreading after 9 days? (Give your answer correct to at least four decimal places.) :1 thousand people per day (d) Compare the function f(t) to the solutions to the different forms of the logistic growth model from this lesson. Enter the differential equation in terms of y that corresponds to this solution. 2V. = dt (e) How many people have heard the news when its rate of spread is 8.64 thousand people per day? (There are two answers Give your answers correct to the nearest whole person.) people (smaller value) people (larger value) (f) At what two times is the news spreading at rate 8.64 thousand people per day? (Give your answers correct to at least three decimal places.) days (smaller value) days (larger value) (g) what is the fastest rate at which the news spreads? (Give your answer correct to at least three decimal places.) thousand people per day Submit Answer 4. [I10 Points] DETAILS MY NOTE! Let y(t) be the percentage of the population that supports a particular government policy. The rate of change of the level of support for the policy is proportional to both the percentage of the population tha agrees with the policy and the percentage that disagrees with the policy. Let the constant of proportionality for this relationship be written as k. (a) Enter the differential equation that is satisfied by y. (Your answer should be given in terms of y. Even though y is a function of time, enter y, rather than y(t).) 1L: dt (b) Suppose that k is positive. Sketch a number of solutions of the differential equation, starting from a range of initial levels of support for the policy. (This will not be graded.) (c) Suppose that k is negative. what do solutions of the differential equation now look like? what happens to the level of support for the policy? (This will not be graded.) (d) Now, suppose that, in addition to people changing their support as described above, people also change from agreeing to disagreeing because they have second thoughts about the policy, Suppo that this rate is proportional to the percentage of people that agree with the policy, with constant of proportionality equal to c Enter the new differential equation that is satisfied by y. EL: dt Submit Answer 5. [-l10 Points] DETAILS MY NOTE: The fish population in a pond with carrying capacity 1500 is modeled by the following logistic equation where N(t) denotes the number of fish at time t in years. Startan at the time at which the number of fish reached 350, the owner of the pond removed (harvested) fish at a Constant rate of 100 fish per year. We take I : 0 to be the time at which the owner Startel to harvest fish from the pond. (a) Modify the differential equation to model the population of fish from the time it reached 350. w dl (b) Sketch several solutions of the new equation, including the solution curve with Mo) = 350, (This will not be graded.) (c) 15 the practice of catching 100 fish per year sustainable or will it totally deplete the fish population in the pond? (You will have only one submission for this answer. Use your submission wiselyl) 0 Yes, it is sustainable. O No, it is unsustainable (d) How far below the original carrying capacity will the number of fish be in the long run? (Give your answer correct to the nearest whole fish.) :isi