Suppose a tank is filled with 85 L of a 45% alcohol solution (by volume). You re

Question

Suppose a tank is filled with 85 L of a 45% alcohol solution (by volume). You repeatedly perform the following operation: Remove 4 L of the solution from the tank and replace them with 4 L of 20% alcohol solution. a. Let C_n be the concentration of the solution in the tank after the nth replacement, where C_0 45%. Write the first five terms of the sequence {C_n}. After how many replacements docs the alcohol concentration reach 25%? Determine the limiting (steady-state) concentration of the solution that is approached after many replacements. The alcohol concentration reaches 25% after 34 replacements. (Round up to the nearest whole number.)

Explanation / Answer

The concentration after 'n'th replacement can be deduced using the following recursive relationship:

Cn+1 = 85L(Cn) -- 4L(Cn) + 4L(20%)/85L

Given : C0 = .45

1) n= 0, C1 = (85* (C0) - 4 * 0.45 + 4*0.2)/85 = 85 * 0.45 -1.8+0.8/85 = 0.4382

2) n=1, C2 = (85*.4382 - 4*0.4382 + 0.8)/85 = 0.4270

3) n=2, C3 = (85*0.4270 - 4*0.4270 + 0.8)/85 = 0.4163

4) n=3, C4 = (85*0.4163-0.4*0.4163+0.8)/85 = 0.4061

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