# Differential Equation Dilution problem

• Dluna08
In summary, the amount of salt in the tank after t min can be expressed as Q = 40 + 4e^-t/10, where t is the time in minutes. To find the amount of salt in the tank when it contains 40 gallons of solution, substitute Q=40 into the expression. This gives us Q=40+4e^-0/10=40+4=44 pounds of salt.
Dluna08
A tank initially holds 80 gal of a brine solution containing 1/8 lb of salt per gallon. At t=0, another brine solution containing 1 lb of salt per gallon is poured into the tank at the rate of 4 gal/min, while the well stirred mixture leaves the tank at the rate of 8 gal/min. a) set-up the DE to find an expression for the amount of salt in the tank after t min. b) find the amount of salt in the tank when the tank contains 40 gallons of solution

## Homework Equations

when i set up my DE i end up with dQ/dt + 1/10Q = 4. I notice that this equation is linear with p(t) = 1/10. So I solve and get Q= 40 + ce^-t/10. I don't know how to find the amount of salt in the tank when the tank contains 40 gallons of solution??

## The Attempt at a Solution

a) dQ/dt + 1/10Q = 4b) Substituting Q=40, we get 0+1/10(40) = 4, which implies that c=4. Thus, the expression for the amount of salt in the tank after t min is: Q = 40 + 4e^-t/10.

## 1. What is a differential equation dilution problem?

A differential equation dilution problem is a mathematical model used to describe the change in concentration of a substance over time due to dilution. It involves solving a differential equation that represents the relationship between the concentration of the substance, the diluting agent, and time.

## 2. How is a differential equation dilution problem solved?

A differential equation dilution problem is typically solved by using analytical or numerical methods. Analytical methods involve finding an exact solution to the differential equation, while numerical methods use computer algorithms to approximate the solution.

## 3. What are the key components of a differential equation dilution problem?

The key components of a differential equation dilution problem include the initial concentration of the substance, the rate of dilution, and the rate of change of concentration over time. These components are represented in the form of a differential equation, which can be solved to determine the concentration of the substance at any given time.

## 4. What are some real-life applications of differential equation dilution problems?

Differential equation dilution problems are used in various fields such as chemistry, biology, and environmental science to model the behavior of substances in solutions. They are also used in industrial processes, such as wastewater treatment, to optimize the dilution process and ensure safe disposal of harmful substances.

## 5. Can differential equation dilution problems be solved for any type of substance?

Yes, differential equation dilution problems can be solved for any type of substance, as long as the relationship between the concentration, diluting agent, and time can be accurately represented by a differential equation. However, the complexity of the problem and the availability of data may affect the accuracy of the solution.

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