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In Problems 25-28 consider the two-compartment model for two tanks with respective volumes $V_1$ and $V_2$. $\frac{dC_1}{dt} = \frac{q}{V_1}(C_\infty - C_1)$ (8.93) $\frac{dC_2}{dt} = \frac{q}{V_2}(C_1 - C_2)$ (8.94) where $C_1(t)$ is the concentration in the first tank and $C_2(t)$ is the concentration in the second tank, and $q$ is the volume of water flowing between the two tanks in one unit of time. 25. When we analyzed (8.93) and (8.94) in the main text we assumed that $V_1 \neq V_2$. Now consider how the analysis must be modified if $V_1 = V_2$, and $C_1(0) = C_2(0) = 0$. a. Show that $C_1(t) = C_\infty(1 - e^{-qt/V_1})$ and $C_2(t) = C_\infty \left(1 - \left(1 + \frac{qt}{V_1}\right)e^{-qt/V_1}\right)$ b. Show that $\lim_{t \to \infty} C_1(t) = C_\infty$ and $\lim_{t \to \infty} C_2(t) = C_\infty$.

          In Problems 25-28 consider the two-compartment model for two tanks with respective volumes $V_1$ and $V_2$.
$\frac{dC_1}{dt} = \frac{q}{V_1}(C_\infty - C_1)$
(8.93)
$\frac{dC_2}{dt} = \frac{q}{V_2}(C_1 - C_2)$
(8.94)
where $C_1(t)$ is the concentration in the first tank and $C_2(t)$ is the concentration in the second tank, and $q$ is the
volume of water flowing between the two tanks in one unit of time.
25. When we analyzed (8.93) and (8.94) in the main text we assumed that $V_1 \neq V_2$. Now consider how
the analysis must be modified if $V_1 = V_2$, and $C_1(0) = C_2(0) = 0$.
a. Show that $C_1(t) = C_\infty(1 - e^{-qt/V_1})$ and $C_2(t) = C_\infty \left(1 - \left(1 + \frac{qt}{V_1}\right)e^{-qt/V_1}\right)$
b. Show that $\lim_{t \to \infty} C_1(t) = C_\infty$ and $\lim_{t \to \infty} C_2(t) = C_\infty$.

In Problems 25-28 consider the two-compartment model for two tanks with respective volumes V1 and V2.
(dC1)/(dt) = (q)/(V1)(C∞ - C1)
(8.93)
(dC2)/(dt) = (q)/(V2)(C1 - C2)
(8.94)
where C1(t) is the concentration in the first tank and C2(t) is the concentration in the second tank, and q is the
volume of water flowing between the two tanks in one unit of time.
25. When we analyzed (8.93) and (8.94) in the main text we assumed that V1 ≠ V2. Now consider how
the analysis must be modified if V1 = V2, and C1(0) = C2(0) = 0.
a. Show that C1(t) = C∞(1 - e^-qt/V1) and C2(t) = C∞(1 - (1 + (qt)/(V1))e^-qt/V1)
b. Show that limt →∞ C1(t) = C∞ and limt →∞ C2(t) = C∞.

Added by Sarah M.

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