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
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