Question: A cascade of several tanks is shown in Fig. Initially, tank 0 contains 1 gal of alcohol and 1 gal of water, while the other

A cascade of several tanks is shown in Fig. 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 O al the rate of I 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 a t time r

(a) Show that x0(t) = e-t/2.
(b) Show by induction that xn(t) = (tne-1/2) / (n!2n). n = 1, 2, ....
(c) Show that the maximum value of xn(t) will be Mn = nne-n /n!.
(d) Use Stirling's approximation, n! ‰ˆ ˆš(2Ï€n) nn e-n to show that Mn ‰ˆ (2Ï€n)-1/2

Tank 0 Tank 1 Tank 2 Tank n

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