1) Two tanks, A and B, are connected to each other via two different pipes. Tank
ID: 3345377 • Letter: 1
Question
1) Two tanks, A and B, are connected to each other via two different pipes. Tank A has 300 gallons of brine, and tank B has 100 Gallons of Brine. Brine flows from tank A to tank B at a rate of 5 gallons/minute. Brine flows from tank B to tank A at a rate of 5 gallons/minute. Each tank initially contains 50 pounds of salt.
a) Write the system of differential equations modeling the above scenario.
b) Solve the system anyway you'd like. (eigenvalues or Laplace transform)
c) How much salt is in the Tank A after 5 minutes?
2) Write the solution to the following differential equation as a convolution of two functions.
y'-y= ln(t)/(t+1) where y(0)=0
Explanation / Answer
CONTINUING BELOW ....
b) Solve the system anyway you'd like. (eigenvalues or Laplace transform)
AS SHOWN ABOVE
Q'=-(Q/60)+0.05R...............................................1
R'=(Q/60)-0.05R.................................................2
DIFFERENTIATING EQN. 1 ....
Q''= -Q'/60 + 0.05R' = -Q'/60 +0.05*Q/60 -0.05*0.05R=
= -Q'/60 +0.05*Q/60 -0.05[Q' + Q/60] = -Q' [ (1/60)+0.05]= -0.06667Q'
PUT Q'=S....
S' = DS/DT = - 2/30 S................................................
DS / S = -2DT/30........................................
LN[S] - LN[C]= -2T/30 .............................................
S = C* [E^(-T/15)] =Q' =DQ/DT ........................
Q = [ - 15 C] [ E^(-T/15)]+K ..........................
AT T=0............Q=50................
50=-15C+K.................K=50+15C
Q= [ - 15 C] [ E^(-T/15)]+ 50+15C..................................................3
Q'= C* [E^(-T/15)]..PUTTING IN EQN. 1 ................
C* [E^(-T/15)] = [-1/60] [ {- 15 C} { E^(-T/15)}+ 50+15C] + 0.05R ...
AT T=0 ....R=50
C= [-1/60] [ {- 15 C} + 50+15C] + 0.05*50 = - 5/6 + 2.5 = 10/6 = 5/ 3
...Q= [ - 25] [ E^(-T/15)]+ 75..................................................4...ANSWER
R = (100/3)* [E^(-T/15)] + [1/3] [ {- 25} { E^(-T/15)}+ 75] .....5 ANSWER
c) How much salt is in the Tank A after 5 minutes?
AT T=5 ......Q= [ - 25] [ E^(-5/15)]+75......ANSWER
LONG LEG WORK
PLEASE POST Q2 SEPARATELY..
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