(a) Reread of Exercises 3.3. In that problem you were

Chapter 4, Problem 27E

(choose chapter or problem)

(a) Reread Problem 8 of Exercises 3.3. In that problem you were asked to show that the system of differential equations

\(\frac{d x_{1}}{d t}=-\frac{1}{50} x_{1}\)

\(\frac{d x_{2}}{d t}=\frac{1}{50} x_{1}-\frac{2}{75} x_{2}\)

\(\frac{d x_{3}}{d t}=\frac{2}{75} x_{2}-\frac{1}{25} x_{3}\)

is a model for the amounts of salt in the connected mixing tanks A, B, and C shown in Figure 3.3.7. Solve the system subject to \(x_{1}(0)=15\), \(x_{2}(t)=10\), \(x_{3}(t)=5\).

(b) Use a CAS to graph \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) in the same coordinate plane (as in Figure 4.9.1) on the interval [0, 200].

(c) Because only pure water is pumped into Tank A, it stands to reason that the salt will eventually be flushed out of all three tanks. Use a root-finding application of a CAS to determine the time when the amount of salt in each tank is less than or equal to 0.5 pound. When will the amounts of salt \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\) be simultaneously less than or equal to 0.5 pound?

Text Transcription:

fracd x_1dt=-frac150x_1

fracd x_2dt=\frac150x_1-frac275x_2

fracd x_3dt=\frac275x_2-frac125x_3

x_1(0)=15

x_2(t)=10

x_3(t)=5

x_1(t)

x_2(t)

x_3(t)