homework 6.docx - 3.1 1 dP/dt = kP P = Poe^kt P(5 = 2Po P(t...

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3.1 1. dP/dt = kP P = Poe^kt P(5) = 2Po k = 1/5 In2 P = ^(In2)t/5 P(t) = 3Po 3 = e^(In2)t/5 In3 = (In2)t/5 t = 5In3/In2 = 7.9 years In4 = (In2)t/5 = t = 10 years 3. P(0) = c = 500 P= 500e^kt P(10) = 500e^10k = 575 k = 1/10 In 575/500 = 10 In 1.15 t=30 P(30) = 500e^(1/10)(In1.15)30 = 500e^3In 1.15 = 760 P’(30) = kP(30) = 1/10 (in 1.15)760 = 10.62 persons/ years 6. 100 97 in 6 hours dA/dt =kA A(t) = Ae^kt 97 = 100e^(kt) In (97/100) = kt In (97/100)/6 = k =-0.005 A(t) = 100e^((_.005)(t)) A(24)= 100e^((-.005)(24)) = 100e^(-.12) = 88.69 mg 13. dT/dt = k(T-10) T = 10 + ce^kt T(0)= 70° T(1/2) = 50° c = 60 k = 2In (2/3) T(1)= 36.67° T(t)= 15° t = 3.06 minutes 15. dT/dt = k(T-100)T = 100 + ce^kt T(0)= 20° T(1)= 22° c = -80 k= In(39/40) T(t) = 90° t= 82.1 seconds T(t)= 98° t= 145.7 seconds 17. dT/dt= k(T-Tm) T(t)= Tm + ce^ktT(0)=70 c= 70-Tm T(t)=Tm+ (70-Tm)e^kt T(1/2) = Tm + (70-Tm)e^k/2 =110 T(1) = Tm + (70-Tm)e^k = 145 E^k = (e^k/2)^2 = (110-Tm / 70-Tm)^2 = 145 – Tm / 70 – Tm, (110 – Tm)^2 / 70 – Tm = 145 -Tm 12100 – 220Tm + T^2 m = 10150 – 250 Tm + T^2 m Tm = 390
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29. L di/dt +Ri = E(t), L = 0.1, R=50, E(t) = 50 i = 3/5 + ce^-500t i(0)= 0 c = -3/5 lin i(t) = 3/5 31. R dq/dt + (1/C)q = E(t), R=200, C= 10^-4, E(t) = 100 q = 1/100 + ce^-50t q(0) = 0 c = -1/100 I = 1/2e^-50t 32. R dq/dt + 1/c q = E(t) e^200t dq/dt + 200e^200t q = 0.2e^200t ∫d.dt (e^200t q) dt = ∫0.2e^200t dt i(0) = 0.4 1000/1000 dq/dt + 1/5(10)^-6 q = 200/100 Dq/dt + 200q = .2 e^200t q = .2/200 ∫200e^200t dt q(t) = .1/1000 + ce^-200t
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