A 400-gallon tank initially contains 100 gallons of brine containing 50 pounds of salt. Brine consisting of 1 pound of salt per gallon enters the tank at the rate of 5 gallons/second, and the well-mixed brine in the tank flows out at the rate of 3 gallons/second. How much salt will the tank contain when it is full of brine?

Question
Asked Sep 14, 2019

A 400-gallon tank initially contains 100 gallons of brine containing 50 pounds of salt. Brine consisting of 1 pound of salt per gallon enters the tank at the rate of 5 gallons/second, and the well-mixed brine in the tank flows out at the rate of 3 gallons/second. How much salt will the tank contain when it is full of brine? 

check_circleExpert Solution
Step 1

Formulate a differential equation as follows. Let A(t) be the rate of change of amount of salt in the tank at time t. Also observe that:

 Rate of change of A = Rate of input – Rate of output.

x3
1x5-100+(5-3)
dt
ЗА
= 5
100 2
ЗА
= 5
dt
100 2
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x3 1x5-100+(5-3) dt ЗА = 5 100 2 ЗА = 5 dt 100 2

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Step 2

Solve the obtained differentia...

dA
ЗА
1002
dt
in 100-2t)
dt
100+2
e'
(1002)
Integrating factor, IF
The general solution is
IFx A(t)5(100+2r) dt
5
(100 2t) A(t) (100 2t)3 C
C
4(t)(100 2) +
|(100 2t)
Apply initial condition A (0) = 50.
С
50 (100 0)
+
+
|(100 0)
C -50000
50000
A()(100 2-
(100 2t)
help_outline

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dA ЗА 1002 dt in 100-2t) dt 100+2 e' (1002) Integrating factor, IF The general solution is IFx A(t)5(100+2r) dt 5 (100 2t) A(t) (100 2t)3 C C 4(t)(100 2) + |(100 2t) Apply initial condition A (0) = 50. С 50 (100 0) + + |(100 0) C -50000 50000 A()(100 2- (100 2t)

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