(a) Solve the di↵erential equation.(b) Plot the family of solutions and include the equilibrium solutions.(c) What happens ast! 1?2. The logistic model has been applied to the natural growth of thehalibut population in certain areas of the Pacific Ocean. Let P, measuredin kilograms, be the total mass (biomass) of the halibut population attime t. The parameters in the logistic equation are estimated to havethe valuesk= 0.71/year andK= 80.5⇥106kg. Let the initial mass beP0= 0.25K.(a) Find the biomass of the halibut after two year.(b) Find the time for whichP(t) = 0.75K.Remark: Check out the DNews youtube video ‘Should We Close The OceanTo Save Fish? ’3. LetN(t) be the population of humans at time t and letPbe thereproduction rate of people. In 1960, Foerster, Mora and Amiotpublished a paper calledDoomsday: Friday November 13th 2026, thatsaid the population of Earth (given it’s advancing technologies) can bemodeled asdN(t)dt=PNwhereP=P0N(t)1k, whereP0andkare constants determined fromhistorical data.(a) Solve the di↵erential equation (absorb the negative in your constantof integration).80
Janelle Resch’s MATH 138 Notes Spring 2014(b) Using historical data, it was found thek= 0.990±0.009 and theproduction rateP0⇡120,000,000 people/year. Sketch the family ofsolutions taking the approximate value forPoandk⇡1.(c) If the carrying capacity of Earth is 9 billion people, in what year mustour population reach an equilibrium so we don’t reach ‘doomsday’?(d) BONUS: According to scientists, if the human population wants toremain as omnivores, Earth can only sustain 2.5 billion people. Whatcan we do in order decrease the population to a sustainable level torectify our over population crisis.Remark: Check out the youtube video ‘immigration by the numbers ’, theBill Nye the Science Guy episode ‘Populations’.81
Janelle Resch’s MATH 138 Notes Spring 2014MATH 138 - Lecture 12Example:Professor Resch has just finished steeping her tea at the perfecttemperature - 120oF. She takes her tea to her office which hasan ambient temperature of 72oF. After 15 minutes, thetemperature of the tea is 110oF.(a)How long will it take before the tea cools down to 90oF?LetT(t) be the temperature of the tea after t minutes.LetTs= 72oFbe the temperature of the office.Lett= 0 be the time at which Professor Resch finishes steepeningher tea and is in her office.Newton’s law of cooling states thatdT(t)dt=-k[T(t)-Ts]dTdt=-k[T-72]Separating the variables and integrating, we obtain1(T-72)dT=-kdtZ1(T-72)dT=-kZdtln|T-72|=-kt+CT(t)-72 =De-ktOur initial condition says that att= 0,T(0) = 120oF, i.e.,T(0)-72 =De-k0120-72 =DD= 48Thus, the solution to our IVP isT(t) = 48e-kt+ 7282
Janelle Resch’s MATH 138 Notes Spring 2014Now let’s solve for k. We know thatT(15) = 110, soT(t= 15) =48e-k15+ 72110-72 =48e-k153848=e-k15-ln(3848)15=kk⇡0.01560554706We want to determine whenT(t) = 90, soT(t) =48e-0.01560554706t+ 7290 =48e-0.01560554706t+ 7218 =48e-0.01560554706t1848=e-0.01560554706tln✓1848◆=-0.01560554706tt⇡62.8737Thus, after about 63 minutes, the temperature of the tea will be90oF.