Tank I in Figure 5 is filled with V1 liters of water...

Tank I in Figure 5 is filled with $V_{1}$ liters of water containing blue dye at an initial concentration of $c_{0}$ g/L. Water flows into the tank at a rate of $R \mathrm{L} / \mathrm{min}$ , is mixed instantaneously with the dye solution, and flows out through the bottom at the same rate $R .$ Let $c_{1}(t)$ be the dye concentration in the tank at time $t$ . $$ \begin{array}{l}{\text { (a) Explain why } c_{1} \text { satisfies the differential equation } \frac{d c_{1}}{d t}=-\frac{R}{V_{1}} c_{1}} \\ {\text { (b) Solve for } c_{1}(t) \text { with } V_{1}=300 \mathrm{L}, R=50, \text { and } c_{0}=10 \mathrm{g} / \mathrm{L} \text { . }}\end{array} $$

Tank I in Figure 5 is filled with $V_{1}$ liters of water containing blue dye at an initial concentration of $c_{0}$ g/L. Water flows into the tank at a rate of $R \mathrm{L} / \mathrm{min}$ , is mixed instantaneously with the dye solution, and flows out through the bottom at the same rate $R .$ Let $c_{1}(t)$ be the dye concentration in the tank at time $t$ .
$$
\begin{array}{l}{\text { (a) Explain why } c_{1} \text { satisfies the differential equation } \frac{d c_{1}}{d t}=-\frac{R}{V_{1}} c_{1}} \\ {\text { (b) Solve for } c_{1}(t) \text { with } V_{1}=300 \mathrm{L}, R=50, \text { and } c_{0}=10 \mathrm{g} / \mathrm{L} \text { . }}\end{array}
$$

Key Concepts

Mixing Problems

Mixing problems involve analyzing how the concentration of a substance changes over time within a system where different fluid streams are combined. In these scenarios, one typically considers both inflows and outflows where the substance concentration may vary, requiring a balance between the addition and removal of the solute to model the system accurately.

Conservation of Mass

The conservation of mass principle is fundamental in mixing problems, dictating that the total mass of the solute in the system changes only by what is added or removed. This principle underpins the formulation of differential equations that account for the solute's accumulation, ensuring that all mass entering or leaving the tank is properly considered.

First-Order Linear Differential Equations

First-order linear differential equations are used to model the rate of change of a quantity that depends linearly on the current state. In mixing problems, the rate at which the concentration changes is often proportional to the current concentration, leading to a linear relationship that can be analyzed using standardized solution techniques.

Separable Differential Equations

Separable differential equations are a class of differential equations in which the variables can be separated on opposite sides of the equation. This method allows for straightforward integration of each side with respect to its variable, often leading to exponential decay or growth solutions in systems such as those encountered in mixing problems.

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