# Differential Equation (mixture problem)

• Tonga
In summary, a beer manufacturer needs to convert 40 gallons of regular beer into light beer by adding water at a rate of 4 gal/min and allowing the mixture to flow out at a rate of 2 gal/min. The light beer has 1/3 the calories of regular beer and is 2/3 water. The formula for calculating the conversion time is da/dt = R1-R2, where R1 is the rate of substance entering and R2 is the rate of substance leaving. It is unclear what concentration should be used in this problem.
Tonga
A beer manufacturer needs to convert a tank containing 40 gallons of regular beer into light beer by adding water. Initially, water is pumped into the tank at the rate of 4 gal/min. And the perfectly mixed solution is allowed to flow out of the tank at the rate of 2 gal/min. The light beer has 1/3 calories of regular beer. That is, it is 2/3 water. How long will it take to convert the regular been into light beer?

Can someone please help me set up and solve this problem. I am confused on what to use as the concentration of the solution.

Formula: da/dt=R1-R2
R1=(rate of substance entering)(concentration)
R2=(rate of substacne leaving)(concentration)

Last edited:
I'm a bit confused by the question... it looks like they should just pump 80 gallons into the tank, then let it all out, because you can't (or can you?) start releasing perfectly mixed beer before you've added enough water to mix it all.

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It is used to describe the relationship between a quantity and its rate of change over time.

## 2. What is a mixture problem in differential equations?

A mixture problem in differential equations involves finding the concentration of a substance in a mixture at a given time, based on the rate at which it is being added or removed from the mixture.

## 3. How do you solve a mixture problem using differential equations?

To solve a mixture problem using differential equations, you must first set up an equation that represents the changing concentration of the substance in the mixture. Then, you can use techniques such as separation of variables or integrating factors to find the solution.

## 4. What are some real-life applications of mixture problems in differential equations?

Mixture problems in differential equations have many applications in fields such as chemistry, biology, and economics. For example, they can be used to model the spread of a disease in a population, the growth of a bacteria culture, or the dilution of a chemical in a solution.

## 5. What are the limitations of using differential equations for mixture problems?

While differential equations can be a powerful tool for solving mixture problems, they may not always provide an accurate representation of real-world scenarios. Assumptions and simplifications made in the modeling process can lead to discrepancies between the predicted and actual results.

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