# Solving a mixing problem with a differential equation

• arhzz
In summary, the conversation discusses a mixing problem where the volume of a tank is modeled using a differential equation. The conversation also explores how to calculate the volume of the tank at any given time and how to determine when the salt content in the tank will reach 15%. The solution involves rearranging the equation to solve for time and using an initial condition.
arhzz
Homework Statement
A tank contains 700 liters of water,to begin the tank is filled with 55 kilograms of salt.35 liters per minute of pure water is flowing in and 35 liters per minute of the water is flowing out.Describe the system using a differential equation (assume the mixing is homogenous).After how many minutes will the salt content be reduced to 15%
Relevant Equations
Differential equations
Hello!

First I tried modelling it like most mixing problems.

$$\frac{dA}{dt} = rate coming in - rate coming out$$ where dA is the volume and dt is the time

rate coming in/out can be describe as; contrencation * flow rate.

Now if we plug that all on

$$\frac{dA}{dt} = 35 * 0 - \frac{A}{700} * 35$$

Now, we can make simplifications,seperate the variables,than integrate and this should come out.

$$ln(A) = \frac{-t}{20} + C$$

Now since a initial condition was given.55 kilograms are in the tank at time 0. we can do this.

A(0) = 55

We should get that C = ln(55);

Plug that back in and use e on both sides gets us too

$$A = 55 e^{\frac{-t}{20}}$$

Now I think this is how can calculate the volume of the tank at any given point t.

Now to get at what point,or after what time salt content will be at 15% I tried the following.

15% of 55 is 8.25 kg,now we know that we want the value of A to be 8.25 but just not when.So we can write it like this.

A(t) = 8.25 kg

If we plug that in;

$$ln(8,25) = 55 e^{\frac{-t}{20}} + C$$

than we can use ln on both sides,multiply by 20 to get rid of the fraction and we should get

t = 37,94 m;

Now I am not sure that this second part is correct.My logic is,I have a formula that will give me the A at any time point,simply reararange to get the time,but I am not sure if it can work that way.Thanks!

Last edited:
Delta2
Yes, you can solve for t in the equation you ended with. I get the same value for t, assuming the differential equation you found is correct, which it seems to be.

Delta2 and arhzz
Perfect thanks!

Delta2

## 1. What is a mixing problem in the context of a differential equation?

A mixing problem refers to a scenario where two or more substances are being combined or mixed together, and the rate at which they are being mixed is changing over time. This can be represented mathematically using a differential equation, which describes the relationship between the rate of change of the mixture and the variables involved.

## 2. How is a mixing problem solved using a differential equation?

To solve a mixing problem with a differential equation, we first need to set up the equation by identifying the variables involved (such as the concentrations of the substances being mixed) and determining their rates of change. We then use appropriate mathematical techniques, such as separation of variables or integrating factors, to solve the differential equation and obtain a general solution. This solution can then be used to find specific values for the variables at different points in time.

## 3. What are some common applications of solving mixing problems with differential equations?

Mixing problems with differential equations have various real-world applications, such as in chemical reactions, pharmaceutical manufacturing, and environmental studies. They can also be used to model biological processes, such as the spread of diseases or the growth of populations, where different substances or entities are interacting and changing over time.

## 4. What are some challenges in solving mixing problems with differential equations?

One of the main challenges in solving mixing problems with differential equations is accurately setting up the equation. This requires a good understanding of the problem and the relationships between the variables involved. Additionally, some mixing problems may have complex equations that are difficult to solve analytically, requiring the use of numerical methods to approximate the solution.

## 5. How can solving mixing problems with differential equations be useful in scientific research?

Solving mixing problems with differential equations allows scientists to model and analyze complex systems and processes, providing insights into their behavior and predicting future outcomes. This can be particularly useful in fields such as chemistry, biology, and environmental science, where understanding the dynamics of mixing and reactions is crucial for developing new technologies and solving real-world problems.

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