Three-Tank Setup Consider the cascading arrangement of tanks shown in Fig. 10, with V1 = 2...

Three-Tank Setup Consider the cascading arrangement of tanks shown in Fig. 10, with V1 = 200 gal,

FIGURE 2.4.10 Three-tank setup for Problem.

V2 = 200 gal, and V3 = 500 gal as the volumes of brine in the three tanks. Each tank initially contains 20 lb of salt. The inflow rates and outflow rates for tanks 1 and 2 are all 5 gal/sec, but the outflow rate for tank 3 is 10 gal/sec. The flow into tanks 1 and 2 is pure water.

(a) Let x(t), y(t), and z(t) be the amounts of salt in tanks 1, 2, and 3, respectively. Set up the differential equation and the initial conditions that describe how the amounts of salt in tanks 1 and 2 are changing at any time t.

(b) Solve for x(t) and y(t).

(c) Using the results of part (b), solve for z(t).