Question

Consider the closed tank system, consisting of three brine tanks. Tank 1 has volume 20 gallons, Tank 2 has volume 10 gallons, and Tank 3 has volume 20 gallons. Brine flows from Tank 1 to Tank 2, from Tank 2 to Tank 3, and from Tank 3 back to Tank 1 at the rate of 20 gallons per minute.

1. a) Letting x1(t), x2(t), x3(t) be the amount of salt in each tank at time t, what is the first order system that x′1, x′2, x′3 satisfies?

2. b) By finding the eigenvalues and eigenvectors of an appropriate matrix, write down the general solution of the first order system in part a).

3. c) Initially there is 20 pounds of salt in Tank 1, and no salt in the other tanks. What is the limiting amount of salt in each tank as t → ∞?

could someone help with this homework question? I got stuck on it for like 3 hours :((( ty for any piece of help!

Answer 1

(2) Consider the closed tank systems, consisting of aree brine tanks. with volumes V₂, V2 and V3. The inflow to tank I is the outflow from tank 3 the inflow to tank 2 is he outflow from tank I and the hat flow to tank three is the outflow from tank 2 , with all three flow rates egnals to r, the resulting system of DES model the weights of the salt in each tank at time t o _ 0MT

Here , v =X3=20 gallows, z togallows and r=20gal 2-to gallons an it 20 10 z = 22013- 13 V V3 X2 = fx * x2 - 20 *4-20 Xz= X1-282 X3 = { X2-4 x3 = 20 x 20 x₂=2x-1(3 V2 V3 the associated system · x'ct)= -1 0 1 Lact. lo 20 - IO eigenvalues of y A = (- l To find = o l - 2 O 2 Tt consider = o det (A -> I) -O -1-7 0 1. 1 27 0 O 2 -1-7T

² -3 -4x²5x = -x (x²+40+5)=0 2_ - [+(2+i) ] [ )+(2-1)]=0 de 20 a 3-2-1 /2 = -2+1 s engenvan For every i we find to own vector To dla0 ArdıI- - 0 11 Il -2 0 / To 2 - -10 To T-2 ola :) Bel60-7B1 [14-2017 TT-2 oto R 1X2 R2 /lolol Lo -2 110, R2162) > R2 I 0 2t lo7 Tolol rio - lol o i le K3~27627 K3 o 1 z o 02 - lo loo olo? X1=X3 , X2= 1 / 2 X3 0,- euste (12)= ( ) - X3=1 gives 2,= (?1 Similarly, v = (1+) and us= (-16) (2) W(t)=ciet art (2e Ut enst uz Note that for Aze, p= (-2), q=1 For the associated eigen reeter we see that 21 23 Then real solus corresponding to the couplex eigen penses e off frost - F.! sint) and De₂ (t) = e 2tr

general son : Plt) = Ciel) + C2 Walt) +3Wz(t) Drit) = (20 tear-lo cost 2oite [ -62 cost-₂ cost) + (6 +asmet citezt "(-62-63) Colett + (C2 (3) (262 tēzt [Q6+243) Cost] Txi(t) 22ct) xalt) / Note that (x1+86 +82 ) (t! = a01+ci+20=50, ire total weight of solute in the tanks in the any give and time is constant, as expected o lim no 120 xit ho

times and in I tank -t and thank-3 in tank-2 - (couch)