Mixture problem (with a twist) Diff. Eq.

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In summary, fresh water is continuously pumped into a 60-gallon tank initially filled with brine at a rate of 3 gallons per minute. The resulting mixture overflows into a second 60-gallon tank also at the same rate, and eventually spills onto the ground. Assuming perfect mixing in both tanks, the water in the second tank will be the saltiest when the concentration in the first tank is equal to the concentration of the original brine. The concentration in the first tank can be found as a function of time, which can then be used to determine the concentration in the second tank.
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1. Beginning at time t=0, fresh water is pumped at the rate of 3 gal/min into a 60-gal tank initially filled with brine. The resulting less-and-less salty mixture overflows at the same rate into a second 60-gal tank that initially contained only pure water, and from there it eventually spills onto the ground. Assuming perfect mixing in both tanks, when will the water in the second tank tast the saltiest. Exactly how salty will it be at this time compared to the original brine?



2. The obvious mixture form... (dA/dt)= (Ri-Ro), but I can't get my brain wrapped around this to start setting it up.



3. The simple mixture problem, with an amt. of liquid * mass of substance being incorporated into a already mixed fluid with a certain concentration, with a rate of change, is simple to set up...but I don't know exactly where to start.
 
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First solve the problem of finding the concentration in the first tank as a function of time. That gives you the Ri for the second tank. Now it's just the usual problem.
 

1. What is a "mixture problem with a twist" in Differential Equations?

A mixture problem with a twist in Differential Equations is a type of problem that involves finding the concentration of a substance in a mixture that is changing over time. The "twist" refers to an additional factor or variable that affects the concentration, such as a chemical reaction or a changing volume of the mixture.

2. How do you set up a differential equation for a mixture problem with a twist?

To set up a differential equation for a mixture problem with a twist, you must first identify the variables involved, such as the initial concentration, the rate at which the mixture is changing, and the rate of the "twist" factor. Then, you can use the general form of a first-order differential equation, dy/dt = ky, where k is the rate constant, and substitute in the relevant variables to create the specific equation for the mixture problem.

3. What are the key steps to solving a mixture problem with a twist using Differential Equations?

The key steps to solving a mixture problem with a twist using Differential Equations are: 1) Setting up the differential equation based on the given information, 2) Solving the differential equation using separation of variables or another appropriate method, 3) Using the solution to find the concentration at a specific time or to create a graph of the concentration over time, and 4) Checking the solution for accuracy and making any necessary adjustments.

4. Can you provide an example of a mixture problem with a twist and its solution using Differential Equations?

Yes, an example of a mixture problem with a twist is a tank initially containing 100 liters of water with a concentration of salt of 0.5 grams per liter. Water with a concentration of salt of 1 gram per liter is poured into the tank at a rate of 2 liters per minute, and the mixture is stirred. The salt concentration is also decreasing at a rate of 0.1 grams per minute due to a chemical reaction. The differential equation for this problem is dC/dt = (2/100)(1-C) - (0.1/100)C, where C represents the concentration of salt. The solution to this equation is C(t) = (10/11)e^(-0.1t) + (1/11), which can be used to find the concentration at any given time or create a graph of the concentration over time.

5. What are some real-world applications of mixture problems with a twist and Differential Equations?

Mixture problems with a twist and Differential Equations have many real-world applications, including determining drug concentrations in the body over time, analyzing chemical reactions in industrial processes, and predicting the spread of pollutants in the environment. They can also be used to model population dynamics, such as the spread of a disease in a population. Essentially, any situation that involves changing concentrations or amounts over time can be modeled using mixture problems with a twist and Differential Equations.

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