A 50-gal tank initially contains 10 gal of fresh water. At t = 0, a brine solution containing 1 Ib of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the tank at the rate of 2 gal/min. Find (a) the amount of time required for overflow to occur

Question

A 50-gal tank initially contains 10 gal of fresh water. At t = 0, a brine solution containing 1 Ib of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well-stirred mixture leaves the tank at the rate of 2 gal/min. Find (a) the amount of time required for overflow to occur and (b) the amount of salt in the tank at the moment of overflow.

Accepted Answer

(a) Here [latex]a=0, b=1, e=4, f=2 ext {, and } V_{0}=10 ext {. }[/latex] The volume of brine in the tank at any time t is given by (7.7) as [latex]V_{0}+e t-f t=10+2 t[/latex] . We require t when [latex]10+2 t=50[/latex] ; hence, t = 20 min.

(b) For this problem, (7.8) becomes

[latex]\frac{d Q}{d t}+\frac{2}{10+2 t} Q=4[/latex]

This is a linear equation; its solution is given in Problem 6.13 as

[latex]Q=\frac{40 t+4 t^{2}+c}{10+2 t}[/latex] (1)

At t = 0, Q = a = 0. Substituting these values into (1), we find that c = 0. We require Q at the moment of overflow, which from part (a) is t = 20. Thus,

[latex]Q=\frac{40(20)+4(20)^{2}}{10+2(20)}=48 lb[/latex]

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Eq. (7.7) : [latex]V_{0}+e t-f t[/latex]

Eq. (7.8) : [latex]\frac{d Q}{d t}+\frac{f}{V_{0}+(e-f) t} Q=b e[/latex]

Question 7.18: A 50-gal tank initially contains 10 gal of fresh water. At t...

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