A multiple cascade is shown in Fig.FIGURE. A multiple cascade.At time t = 0, lank 0 contai...

A multiple cascade is shown in Fig.

FIGURE. A multiple cascade.

At time t = 0, lank 0 contains 1 gal of ethanol and 1 gal of water; all the remaining tanks contain 2 gal of pure water each. Pure water is pumped into tank 0 at 1 gal/min, and the varying mixture in each tank is pumped into the mixtures are kept perfectly uniform by stirring. Let x„(t) denote the amount of ethanol in tank n at time t.

(a) Show that x0(t) − e−t/2.

(b) Show by induction on n that

(c) Show that the maximum value of x„(t) for n > 0 is Mn = xn (2n) = nnen/n!.

(d) Conclude from Stirling’s approximation