A tank initially holds 100 gal of a brine solution containing 1 Ib of salt. At t = 0 another brine solution containing 1 Ib of salt per gallon is poured into the tank at the rate of 3 gal/min, while the well-stirred mixture leaves the tank at the same rate. Find (a) the amount of salt in the tank

Question

A tank initially holds 100 gal of a brine solution containing 1 Ib of salt. At t = 0 another brine solution containing 1 Ib of salt per gallon is poured into the tank at the rate of 3 gal/min, while the well-stirred mixture leaves the tank at the same rate. Find (a) the amount of salt in the tank at any time t and (b) the time at which the mixture in the tank contains 2 Ib of salt.

Accepted Answer

(a) Here [latex]V_{0}=100, a=1, b=1 ext {, and } e=f=3 ext {; }[/latex] hence, (7.8) becomes

[latex]\frac{d Q}{d t}+0.03 Q=3[/latex]

The solution to this linear differential equation is

[latex]Q=c e^{-0.03 t}+100[/latex] (1)

At t = 0, Q = a=1. Substituting these values into (1), we find [latex]1=c e^{0}+100, ext { or } c=-99[/latex] . Then (1) can be rewritten as

[latex]Q=-99 e^{-0.03 t}+100[/latex] (2)

(b) We require t when Q = 2. Substituting Q = 2 into (2), we obtain

[latex]2=-99 e^{-0.03 t}+100 \quad ext { or } \quad e^{-0.03 t}=\frac{98}{99}[/latex]

from which

[latex]t=-\frac{1}{0.03} \ln \frac{98}{99}=0.338 min[/latex]

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

Question 7.17: A tank initially holds 100 gal of a brine solution containin...

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