Three large tanks contain brine, as shown below. Use the information in the figu

Question

Three large tanks contain brine, as shown below. Use the information in the figure to (A) x,(a), x(t), x, () at time t in tanks A, B, and C, respectively for the number of pounds of salt Without solving the system, (B) predictlimitingvaluof x. (1), x,(1), x,(t) at t PART 2 (C) Solve the system found in part (A) subject to ()=15, 0)-10, x(0)= 5 (D) Use Maple to graph x(), x(, andx,() in the same coordinate plane on the interval [0, 200] (E) Because only pure water is pumped into Tank A, it stands to reason that the salt will eventually be flushed out of all three tanks. Use Maple or your graphing calculator to determine the time when the amount of salt in each tank is less than or equal to 0.5 pound. When will the amounts of salt (), x(), and,(t) be simultaneously less than or equal to 0.5 pound? ANSW ER -PART -2- GIVEN 50 dt 50 75 dt 75 25

Explanation / Answer

Integrating x_1 gives

x_1= A exp(-t/50)

HEnce limiting value of x_1=0 as the argument of the exponential is negative for all t>0

x_2+2x_2/75= A exp(-t/50)/50

Integrating factor is exp(2t/75)

(x_2'+2x_2/75) exp(2t/75)= A exp(t(-1/50+2/75)/50\

(x_2 exp(2t/75))' = A exp(-t/150)/50\

x_2 exp(2t/75)= -A exp(-t/150)(50*1/150)+B=-3A exp(-t/150)+B

x_2= -3A exp(-t/50)+B exp(-2t/75)

Again the argument of exponential is negative hence limiting value of x_2 is 0

x_3'=-6A exp(-t/50)/75+2B exp(-2t/75)/75-x_3/25

x_3'+x_3/25=-6A exp(-t/50)/75+2B exp(-2t/75)/75

Integrating factor is exp(t/25)

Multiplying gives

(x_3'+x_3/25) exp(t/25)=-6A exp (t/50)/75+2B exp(t/75)/75

Integrating gives

x_3 exp(t/25)= -4A exp(t/50)+2B exp(t/75)+C

x_3=-4A exp(-t/50)+2B exp(-2t/75)+C exp(-t/25)

Again the arguments of all exponential terms are negative for all t>0 hence limiting value of x_3=0

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