cheatham (sc36975) – HW05 – um – (53890)1Thisprint-outshouldhave15questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.001 (part 1 of 3) 10.0 pointsAn initial deposit of $Pis made into anaccount that earns 8% interest compoundedcontinuously. Money is then withdrawn at aconstant rate of $5000 per year.Set up the differential equation for theamountA=A(t) (in thousands of dollars)in the account aftertyears.1.dAdt= 8A−52.dAdt= 8A+ 53.dAdt=A−54.dAdt= 0.08A−5correct5.dAdt= 0.08A+ 5Explanation:The interest earned by the account will in-crease the balance at a rate of 0.08Awhile theconstant withdrawals of $5000 will reduce thebalance at a constant rate 5, the account bal-anceA(t), expressed in thousands of dollars,will satisfy the equationdAdt= 0.08A−5.002 (part 2 of 3) 10.0 pointsSolve the differential equation in part 1.1.A(t) = 62.5 +P1000+ 62.5e0.08t2.A(t) = 62.5 +62.5−P1000e0.08t3.A(t) = 62.5 +P1000−62.5e0.08tcor-rect4.A(t) = 0.625 +P1000−0.625e0.08t5.A(t) =P1000−62.5e0.08tExplanation:After rewriting the differential equation inpart 1 asdAdt−0.08A=−5,it can be solved by using the integrating factorμ(t) =e−0.08dt=e−0.08t.In this casee−0.08tdAdt−0.08Ae−0.08t=ddtAe−0.08t=−5e−0.08t,soA(t) =e0.08tC+ 62.5e−0.08t=Ce0.08t+ 62.5.withCan arbitrary constant.The initialdeposit determinesCsinceA(0) =P1000=⇒C=P1000−62.5.ThusA(t) = 62.5 +P1000−62.5e0.08t.003 (part 3 of 3) 10.0 pointsIf the account balance becomes zero after8 years, what was the amount of the initialdeposit $P? (This value ofPis often calledthePresent Valueof the regular withdrawals$5000 over a period ofnyears.)1.$34416
