# Calculus 2- Differential Equation Mixing Problem

• 1LastTry
In summary, the tank is adding salt at a rate of 2 L/min and the solution is constantly being mixed and evacuated. The salt is being removed at a rate of 0.06 kg/min.
1LastTry

## Homework Statement

A tank containing 400 liters of water has 10 kg of salt solute (dissolved salt).
Some brine containing 0.03kg/L of salt is then introduced at a rate of 2 L/min.
The solution is constantly mixed and evacuated at a rate of 2L/min, such that the volume remains constant. If Q(t) is deﬁned as the quantity of salt (in kg) dissolved in the tank after time t (in minutes),

How do you find the differential equation?

dQ/dt = Rin-Rout

## The Attempt at a Solution

I have no idea how to find Rin or Rout, what do you need in their equations?

1LastTry said:
I have no idea how to find Rin or Rout, what do you need in their equations?

Surely you know how much salt is being added! That information is given to you directly.

If Q(t) is the amount of salt in the tank, the volume of the solution is 400 L and the rate at which the solution is being removed is 2 L/min, then how much salt is being removed every minute?

Rin is added and Rout is removed right?

ummmm 0.03kg/l, so 0.06 kg of salt is being removed every minute?

0.03kg/l introduced at 2l/min isn't it also 0.06 being introduced? or do i have to add 10kg of solute with it?

I am confused with these kind of word problems and my english is bad.

1LastTry said:
ummmm 0.03kg/l, so 0.06 kg of salt is being removed every minute?

How can you know how much salt is being removed, without knowing how much salt there is?

writing equations involving unknown quantities is the whole point of algebra, isn't it?

A tank containing 400 liters of water has 10 kg of salt solute (dissolved salt).
Some brine containing 0.03kg/L of salt is then introduced at a rate of 2 L/min.
The solution is constantly mixed and evacuated at a rate of 2L/min, such that the volume remains constant.
Q(t) is the amount of salt in the tank so Q(t)/400 is the amount of salt per Liter. Since the volume remains constant, brine is going out at 2 L/min, the same rate it is coming in, and that means 2(Q(t)/400)= Q(t)/200. dQ/dt represents the change in Q and there are two kinds of change: salt coming in will be positive, salt going out will be negative.

## What is a differential equation mixing problem?

A differential equation mixing problem involves finding the concentration of a substance in a solution over time, taking into account factors such as rate of flow and rate of change.

## What is the purpose of solving differential equation mixing problems?

The purpose of solving these types of problems is to understand how substances mix and change over time, which is important in fields such as chemical engineering, environmental science, and pharmacology.

## What are the steps to solving a differential equation mixing problem?

The first step is to set up the differential equation by identifying the variables and their rates of change. Then, solve the equation using separation of variables or other methods. Finally, use the solution to answer the specific question about the mixing problem.

## What are some common applications of differential equation mixing problems?

These types of problems are commonly used in real-world scenarios, such as predicting the spread of pollution in a river, determining the concentration of a drug in a patient's bloodstream, or designing a chemical reaction with specific reaction rates.

## Are there any limitations to using differential equations to model mixing problems?

While differential equations can provide accurate solutions to mixing problems, they may not account for all factors and can sometimes be simplified or approximated. Additionally, the accuracy of the solution depends on the accuracy of the initial conditions and the assumptions made in the model.

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