# A solve the di erential equation b plot the family of

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(a) Solve the di erential equation. (b) Plot the family of solutions and include the equilibrium solutions. (c) What happens as t ! 1 ? 2. The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. Let P, measured in kilograms, be the total mass (biomass) of the halibut population at time t. The parameters in the logistic equation are estimated to have the values k = 0 . 71 / year and K = 80 . 5 10 6 kg. Let the initial mass be P 0 = 0 . 25 K . (a) Find the biomass of the halibut after two year. (b) Find the time for which P ( t ) = 0 . 75 K . Remark: Check out the DNews youtube video ‘Should We Close The Ocean To Save Fish? ’ 3. Let N ( t ) be the population of humans at time t and let P be the reproduction rate of people. In 1960, Foerster, Mora and Amiot published a paper called Doomsday: Friday November 13th 2026 , that said the population of Earth (given it’s advancing technologies) can be modeled as d N ( t ) d t = PN where P = P 0 N ( t ) 1 k , where P 0 and k are constants determined from historical data. (a) Solve the di erential equation (absorb the negative in your constant of integration). 80
Janelle Resch’s MATH 138 Notes Spring 2014 (b) Using historical data, it was found the k = 0 . 990 ± 0 . 009 and the production rate P 0 120 , 000 , 000 people/year. Sketch the family of solutions taking the approximate value for P o and k 1. (c) If the carrying capacity of Earth is 9 billion people, in what year must our population reach an equilibrium so we don’t reach ‘doomsday’? (d) BONUS: According to scientists, if the human population wants to remain as omnivores, Earth can only sustain 2.5 billion people. What can we do in order decrease the population to a sustainable level to rectify our over population crisis. Remark: Check out the youtube video ‘immigration by the numbers ’, the Bill Nye the Science Guy episode ‘Populations’. 81
Janelle Resch’s MATH 138 Notes Spring 2014 MATH 138 - Lecture 12 Example: Professor Resch has just finished steeping her tea at the perfect temperature - 120 o F . She takes her tea to her o ffi ce which has an ambient temperature of 72 o F . After 15 minutes, the temperature of the tea is 110 o F . (a) How long will it take before the tea cools down to 90 o F ? Let T ( t ) be the temperature of the tea after t minutes. Let T s = 72 o F be the temperature of the o ffi ce. Let t = 0 be the time at which Professor Resch finishes steepening her tea and is in her o ffi ce. Newton’s law of cooling states that d T ( t ) d t = - k [ T ( t ) - T s ] d T d t = - k [ T - 72] Separating the variables and integrating, we obtain 1 ( T - 72) d T = - k d t Z 1 ( T - 72) d T = - k Z d t ln | T - 72 | = - kt + C T ( t ) - 72 = De - kt Our initial condition says that at t = 0, T (0) = 120 o F , i.e., T (0) - 72 = De - k 0 120 - 72 = D D = 48 Thus, the solution to our IVP is T ( t ) = 48 e - kt + 72 82
Janelle Resch’s MATH 138 Notes Spring 2014 Now let’s solve for k. We know that T (15) = 110, so T ( t = 15) =48 e - k 15 + 72 110 - 72 =48 e - k 15 38 48 = e - k 15 - ln ( 38 48 ) 15 = k k 0 . 01560554706 We want to determine when T ( t ) = 90, so T ( t ) =48 e - 0 . 01560554706 t + 72 90 =48 e - 0 . 01560554706 t + 72 18 =48 e - 0 . 01560554706 t 18 48 = e - 0 . 01560554706 t ln 18 48 = - 0 . 01560554706 t t 62 . 8737 Thus, after about 63 minutes, the temperature of the tea will be 90 o F .
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