Question

A tank initially contains 100 gal of pure water. Starting at \(t=0\), a brine containing \(4 \mathrm{lb}\) of salt per gallon flows into the tank at the rate of \(5 \mathrm{gal} / \mathrm{min}\). The mixture is kept uniform by stirring and the well-stirred mixture flows out at the slower rate of \(3 \mathrm{gal} / \mathrm{min}\). (a) How much salt is in the tank at the end of \(20 \mathrm{~min} ?\) (b) When is there \(50 \mathrm{lb}\) of salt in the tank?

Short answer

\(dS/dt = 20 - \frac{3S(t)}{100 + 2t}\) (a) After solving the differential equation, we find that S(20) = 40 lb of salt in the tank at the end of 20 minutes. (b) To find when there is 50 lb of salt in the tank, we need to solve for t when S(t) = 50. This occurs at approximately \(t \approx 29.71\) minutes.

Step-by-step solution

Model the saltwater system.

Let S(t) be the amount of salt in the tank at time t. The initial condition is S(0) = 0. The brine is entering the tank at a rate of 5 gal/min with a salt concentration of 4 lb/gal, so the inflow salt rate is 5 * 4 = 20 lb/min. The well-stirred mixture flows out at the slower rate of 3 gal/min. Since the tank is being stirred, the salt concentration of the mixture in the tank is uniform. Thus, the outflow rate of salt is (3 * S(t)) / V(t), where V(t) is the volume of the mixture at time t. The volume equation is V(0) = 100 and V(t) = 100 + 5t - 3t = 100 + 2t. From this, we can model the differential equation for dS/dt.

Key concepts

Saltwater Tank Problem

The Saltwater Tank Problem involves understanding how the concentration of salt changes over time in a tank. Imagine you start with a tank containing 100 gallons of pure water. Brine, a salty solution, flows into this tank, bringing salt along with it. This flow is constant at 5 gallons per minute, with a concentration of 4 pounds of salt per gallon. The tank is well-stirred, ensuring the salt mixes evenly. Meanwhile, the mixture drains out at 3 gallons per minute.

This creates a dynamic situation where salt enters and leaves the tank. To determine the amount of salt at any given moment, we need to carefully model this system using mathematical tools, accounting for rates of inflow and outflow. This understanding lays the groundwork for solving further questions, like the amount of salt after a specific time or the time when a certain salt concentration is reached.

Initial Value Problem

An Initial Value Problem (IVP) in differential equations gives us a starting point to work with. Here, our IVP begins with knowing that at time zero (\(t=0\)), the amount of salt in the tank is zero (\(S(0) = 0\)). This initial condition is crucial because it anchors our equation in a real-world scenario.

As the problem progresses, knowing how much salt you start with helps track the changes over time. IVPs often accompany differential equations to specify a unique solution, guiding us to predict the system's behavior accurately based on the starting state.

Differential Equation Modeling

Differential Equation Modeling involves setting up mathematical equations to describe how systems change. In our case, we model how the salt concentration in the tank changes with time. We define \(S(t)\) as the amount of salt in the tank at any time \(t\), and set up a differential equation to represent the rate of change of salt.

This equation considers:

• The inflow of salt: 20 pounds per minute (since 5 gallons of brine with 4 pounds per gallon enter the tank).
• The outflow of salt: depends on the concentration in the tank, calculated as \((3 \times S(t)) / V(t)\), where \(V(t) = 100 + 2t\).

By solving this model, you can predict the salt amount present at any given time, offering insights into how systems in engineering and natural processes behave over time.

Separable Differential Equations

Separable Differential Equations are a powerful tool in solving real-world problems like ours. These equations allow us to separate variables, making complex equations more manageable. In our saltwater tank scenario, we start with a differential equation that describes how salt quantity changes over time.

This equation takes the form \(\frac{dS}{dt} = \text{inflow rate} - \text{outflow rate}\). Separable means we can rearrange it to isolate \(S(t)\) and \(t\) on opposite sides of the equation. This separation allows the integration of both sides, leading to solutions that describe how salt accumulates or depletes over time.

By applying this method, we break down the problem into simpler steps, helping predict outcomes like when the tank will reach 50 pounds of salt.