Question

Suppose that a large mixing tank initially holds 400 gallons of water in which 65 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 6 gal/min, and when the solution is well stirred, it is pumped out at a slower rate of 5 gal/min. If the concentration of the solution entering is 3 lb/gal, find the amount of salt in the tank after 10 minutes.

PLEASE SHOW ALL WORK AND USE DIFF. EQs. THANKS!

Answer 1

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salt m the ut yet) be the tank at time 't ampunt Fin amount that Rlowurd mto the tank Hate) out ameunt that Bloured out of tank. Chate) yin-yout dt ) yun =toncem hatien of salt in inpreo nut uate brine yein-31b/gal)(6gal/min) 18 1b/muin a slour ate Solution is pumped out Solution is accumulating at the rate (6-5) gat/min at 1 min al gM t min s tgal t munuty hre uill b After (400+ t) gallons im tank (toneenhation in out plou C)bga tS gallmin saut eutrut rate Your 400+t 5 mn (4041) your

Fuom ,2) and (3 dy dt in-yout 1 dy 18 (S dt dy + S4 18 dt whis is inear ditteruntial equation dt Snl400tt) 5 Ame nfietre If e : e 5Lne 5 (ttho5 Ct+400) auy Ctt400) + 18 Ct4400)5 dt 4 dy Ctt400)+ (t+40) 5y 18 t+4o0) 7 dt dt at 18 Ctt400) c'. 6

3Ct 4o0)C Snitially theru uas 65 pounas of Salt dissolmed YCo) 65. 66 LOR4 -S 3 Ct+4o0) + C[t+400 4Co)=65 -5 65= 3(400) + c [400 65=1200 + C 5 (4o) F C -454Ox 1o -1135 (400)5 CE -454 x10" yctt4oo3Ct+ 4o 454x 10" After Lo munty CIo) 0410 0 30410-454x 1o") 30410) - 454 x 10 (41) 5 105 12 30- 3.918 X10x 10 -1230-3.918 226 082 pounds of salt