Question

A tank originally contains 100 gallons of fresh water. Water containing lb of salt per gallon is poured into the tank at a rate of 2 gallons per minute, and the mixture is allowed to leave at the same rate. After 10 minutes the salt water solution flowing into the tank suddenly switches to fresh water flowing in at a rate of 2 gallons per minute, while the solution continues to leave the tank at the same rate (a) Write down the initial-value problem that models the amount of salt in the tank b) Solve the initial-value problem using Laplace transforms

Answer 1

Question: tank osiginally contains loo gallons o fsesh watexwates Containing b of Salt pex gallons iS poused into the tank at a gate of agau per minuteand the mdise allowed at the Sama sate. Af士 lo mine the sout uctes galution 위ewing into the tank Suddenly Bwithes to fsest watu Flowing in a sate of Bolutions ontincus to leave the tank at Sane sate gallons pa mnute sle the t doun the initial value psoblem that medel the amount of Salt in the tank Bove the inthal value pRoblems uSing laplace Let Q(t) denote 士he amount of 8 tank at any time t. fos the fiast lornins, ce the pao e83, ±he ton) ditfesntial equation wil be oasesponding ec60 10o 50

50 The Initial dend; ti cnis al。) -o laplace tsansfo2m cn both sides cf. eauatien L50 since y'3-Sys S t By pa*tial fsactiens, 1 50 p, .50 50 50 Take invease lopla e tsansfos 5+1 -t

hese foxe the onount of Salt the tankig }η fAst to minutes, the amount cf Salt in e tank is -Iolso -/Ks This is the initial amount e Sat ios the hext paut o the paccess Atte ten eauation the cossesponcng dfPent to0 5D da dt 50 By Sepasting the vaiables 50

Integsating 50 where c ec. To find c , use the initial Condttion Q() -1s 50 1-e hese foae the amount oe 9ait at time at any time t>o is The concentsation of 9alt lomin atter the fsest, 씨ata 8tooted to Pous into the tank is I0