1. A 120-gal tank contains 90 lb of salt dissolved in 90 gal of water. Brine con

Question

1. A 120-gal tank contains 90 lb of salt dissolved in 90 gal of water. Brine containing 2 lb/gal of salt flows into the tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 33 gal/min
a. find a function x(t) that gives the anount of salt in the tank at time t
b. at what time t will the tank fill with liquid?
c. use the function from part (a) and the answer to part (b) to determine the amount of salt that the tank contains when it is full
d. what is the concentration of salt in the tank when it is full of the liquid?

part B of the qn
consider a cascade of two tanks with V1=100gal and V2=200gal representing the volumes of brine in the tanks. Each tanks also initially contains 50 lb of salt. Pure water flows into tank 1 at the rate of 5 gal/min and the well-stirred mixture leaves the tank 1 and enters the directly into tanks 2 at the same rate of 5 gal/min. a. find the amount of salt in tank 1 at time t b. find the amount of salt in tank 2 at time t c. what is the maximum amount os salt that is possible in tank 2 1. A 120-gal tank contains 90 lb of salt dissolved in 90 gal of water. Brine containing 2 lb/gal of salt flows into the tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 33 gal/min
a. find a function x(t) that gives the anount of salt in the tank at time t
b. at what time t will the tank fill with liquid?
c. use the function from part (a) and the answer to part (b) to determine the amount of salt that the tank contains when it is full
d. what is the concentration of salt in the tank when it is full of the liquid?

part B of the qn
consider a cascade of two tanks with V1=100gal and V2=200gal representing the volumes of brine in the tanks. Each tanks also initially contains 50 lb of salt. Pure water flows into tank 1 at the rate of 5 gal/min and the well-stirred mixture leaves the tank 1 and enters the directly into tanks 2 at the same rate of 5 gal/min. a. find the amount of salt in tank 1 at time t b. find the amount of salt in tank 2 at time t c. what is the maximum amount os salt that is possible in tank 2

Explanation / Answer

Since we have 4gal/min *2lb/gal, every minute 4*2=8 lbs flows in. Since we assume the mixture is well stirred, every minute 33gal/min * x(t)/(90-29t) lbs flows out. So we have the DE x'(t)= 8- x(t)/(90-29t) where x(0)=90 So we have a first order linear DE x'(t)+x(t)/(90-29t)=8 I'm running out of time here so I assume you know how to solve these types of DE's (multiplying by the integrating factor e^(int 1/(90-29t) dt makes it exact).

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