Mathematics, 26.06.2020 16:01 Isaiahtate053
A tank contains 70 lb of salt dissolved in 400 gallons of water. A brine solution is pumped into the tank at a rate of 4 gal/min; it mixes with the solution there, and then the mixture is pumped out at a rate of 4 gal/min. Determine A(t), the amount of salt in the tank at time t, if the concentration of salt in the inflow is variable and given by cin(t)
Answers: 1
![A(t)=400C_{in}(t)+[70-400C_{in}(t)]\cdot e^{-\frac{t}{100}}](/tpl/images/0694/7644/a7505.png)
![\int(Ae^{\frac{t}{100}})'=\int [4C_{in}(t)e^{\frac{t}{100}}]dt\\Ae^{\frac{t}{100}}=4*100C_{in}(t)e^{\frac{t}{100}}+C, $(C a constant of integration)\\Ae^{\frac{t}{100}}=400C_{in}(t)e^{\frac{t}{100}}+C\\$Divide all through by e^{\frac{t}{100}}\\A(t)=400C_{in}(t)+Ce^{-\frac{t}{100}}](/tpl/images/0694/7644/94e7f.png)
![A(t)=400C_{in}(t)+Ce^{-\frac{t}{100}}\\70=400C_{in}(t)+Ce^{-\frac{0}{100}}\\C=70-400C_{in}(t)\\$Therefore, the amount of salt in the tank at any time t is:\\\\A(t)=400C_{in}(t)+[70-400C_{in}(t)]\cdot e^{-\frac{t}{100}}](/tpl/images/0694/7644/a82e6.png)
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A tank contains 70 lb of salt dissolved in 400 gallons of water. A brine solution is pumped into the...
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