# 1 construct the di ff erential equation for q t 2

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(1) Construct the differential equation forQ(t).(2) Solve the differential equation.(3) Predict the long term quantity of the salt in the reservoir.
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Chapter 6 / Exercise 8
Trigonometry
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7.4 Exponential Growth and DecayIf the population growthdPdtis proportional to the population sizeP(t), thendPdt=kP.The solution isP(t) =P(0)ekt.k >0growth,k <0decay.Absolute growth rate = rate of change of population =dP/dt,Relative growth rate = percent change per unit time = (dP/dt)/P,Doubling time (D) of exponential growth = the time required for it to double:P(t) =P(0)2t/D.Half life (H) of exponential growth = the time required for it to be half:P(t) =P(0)12t/H.Newtons Law of Cooling states that the rate of change of the temperature of a coolingbody is proportional to the difference between its temperature T and the temperatureof its surrounding mediumTs. The model is:dTdt=-k(T-Ts),wherekis a constant.Example.A bacteria culture growth at a rate proportional to its size. After 2 hours thereare 40 bacteria and after 4 hours the count is 120. Find an expression for the populationafter t hours.
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Note thatP(2) = 40 andP(4) = 120, we obtain40 =P(0)e2k,120 =P(0)e4k.These imply thatP(0) =403ande2k= 3,ork= ln 3/2.We thus haveP(t) =4033t/2=4033t=403e(ln 3/2)t.Example.The half-life of Sodium-24 is 15 hours. Suppose you have 100 grams of Sodium-24. How many grams remaining after 27 minutes (keep three decimals)?
Example.Use Newton’s Law of Cooling to determine the time of death of a healthy man.He died in his room some time before noon;At noon, his body temperature was found to be 70 degrees;His body cooled another 5 degrees in 1 hour after noon;The room temperature was a constant 60 degrees;Normal temperature of people’s body is 98.6 degree.
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FromT(0) = 70 we get 70 = 60 +C,C= 10 andT(t) = 60 + 10e-kt.At 1:00pm, his body temperature is 70-5 = 65. HenceT(1) = 65.65 = 60 + 10e-k(1),5 = 10e-k,k= ln 2,T(t) = 60 + 10e-tln 2.Thus98.6 = 60 + 10e-tln 2,t=-ln 3.86/ln 2 =-1.95,which means 1 hour and 0.95(60) minutes before noon, or 1 hour and 57 minutes beforenoon. Or the time of death is 10:03AM.

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Term
Winter
Professor
Kousha
Tags
dx, lim P, example
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The document you are viewing contains questions related to this textbook. The document you are viewing contains questions related to this textbook.
Chapter 6 / Exercise 8
Trigonometry
Larson Expert Verified
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