Changing Midstream A 1,000-gallon tank contains 200 gal of pure water. (See Fig. 6.) A bri...

Changing Midstream A 1,000-gallon tank contains 200 gal of pure water. (See Fig. 6.) A brine solution containing 1 lb of salt per gal is flowing into the tank at a rate of 4 gal/sec, and the well-stirred mixture is leaving the tank at the same rate. Let x denote the amount of salt in the tank at time t.

(a) Set up (but do not solve) the initial-value problem (both DE and initial condition).

Figure 6 Tank setup for Problem.

(b) Suppose that this situation continues for a very long time. What is the equilibrium solution xeq?

(c) After the solution has reached equilibrium, an additional faucet is turned on that supplies brine containing 2 lb/gal at a rate of 2 gal/sec. (See Fig. 7.) Set up the new initial-value problem, assuming that the clock is restarted when the new faucet is turned on.

FIGURE 2.4.7 Tank setup for Problem with a second faucet.

(d) How long tf does it take for the tank to fill completely after the second faucet is turned on?

(e) How much salt is in the tank when t = tf?

(f) Set up the IVP for the amount of salt in the tank for t > tf, assuming that the tank overflows.