More Cascading Tanks A cascade of several tanks is shown in Fig. 9. Initially, tank 0 cont...

More Cascading Tanks A cascade of several tanks is shown in Fig. 9. Initially, tank 0 contains 1 gal of alcohol and 1 gal of water, while the other tanks contain 2 gal of pure water. Fresh water is pumped into tank 0 at the rate of 1 gal/min, and the varying mixture in each tank is pumped into the next tank at the same rate. Let xn(t) denote the amount of alcohol in tank n at time t.

FIGURE 2.4.9 Cascading tanks for Problem.

(a) Show that x0(t) = e−t/2.

(b) Show by induction that xn(t) = (tne−t/2)/(n! 2n), n = 1, 2, ….

(c) Show that the maximum value of xn(t) will be Mn = nn e−n/n!.

(d) Use Stirling’s approximation, , to show that Mn ≈ (2πn)−1/2.