cheatham (sc36975) – HW05 – um – (53890)1 Thisprint-outshouldhave15questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 3) 10.0 points An initial deposit of $Pis made into an account that earns 8% interest compounded continuously. Money is then withdrawn at a constant rate of $5000 per year. Set up the differential equation for the amountA=A(t) (in thousands of dollars) in the account aftertyears. 1.dAdt = 8A−5 2.dA dt = 8A+ 5 3.dA dt =A−5 4.dA dt = 0.08A−5correct 5.dA dt = 0.08A+ 5Explanation: The interest earned by the account will in- crease the balance at a rate of 0.08Awhile the constant withdrawals of $5000 will reduce the balance at a constant rate 5, the account bal- anceA(t), expressed in thousands of dollars, will satisfy the equation dA dt = 0.08A−5. 002 (part 2 of 3) 10.0 points Solve the differential equation in part 1. 1.A(t) = 62.5 +P 1000 + 62.5e 0.08t 2.A(t) = 62.5 +62.5−P 1000 e 0.08t 3.A(t) = 62.5 +P 1000 −62.5e0.08t cor- rect4.A(t) = 0.625 +P 1000 −0.625e 0.08t 5.A(t) =P 1000 −62.5e 0.08t Explanation: After rewriting the differential equation in part 1 as dA dt −0.08A=−5, it can be solved by using the integrating factor μ(t) =e−0.08dt =e−0.08t . In this case e−0.08tdA dt −0.08Ae −0.08t =d dt Ae−0.08t=−5e−0.08t , so A(t) =e0.08tC+ 62.5e −0.08t =Ce0.08t + 62.5. withCan arbitrary constant.The initial deposit determinesCsince A(0) =P1000=⇒C=P 1000 −62.5. Thus A(t) = 62.5 +P 1000 −62.5e0.08t . 003 (part 3 of 3) 10.0 points If the account balance becomes zero after 8 years, what was the amount of the initial deposit $P? (This value ofPis often called thePresent Valueof the regular withdrawals $5000 over a period ofnyears.) 1.$34416

cheatham (sc36975) – HW05 – um – (53890)2 2.$36576 3.$26799 4.$29544correct 5.$32077 Explanation: Since the balance becomes zero after 8 years, 0 = 62.5 +P 1000 −62.5e0.64 . Solving forPwe see that P= 62500 1−e−0.64 ∼$29544. 00410.0 points Diana is sick in hospital with a severe bac- terial infection.She is to be fed antibiotics intravenously at a constant infusion rate, and the doctor knows that the anti-biotic will be eliminated from the bloodstream at a rate proportional to the amounty(t) present in the bloodstream at timet. If the half-life of the anti-biotic in the bloodstream is 2 hours, what rate of infusion should he prescribe to maintain a long-term amount of 390 mgs in

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