# Mixing Tank, Differential Equations Problem

• bdh2991
In summary, a larg tank is filled with 500gals of water with 400lbs of salt. The salt is pumped in at a rate of 3gal/min and the solution is pumped out at a rate of 7 gal/min. I tried to find the differential of the equation but I keep getting more salt than when I started. Please help.
bdh2991

## Homework Statement

a larg tank is filled with 500 gals of water with 400lbs of salt. pure water is pumped into the tank at a rate of 3gal/min. the well mixed solution is pumped out at a rate of 7 gal/min.

I need help finding the Differential

## The Attempt at a Solution

I tried coming up with the D.E. and this is what i got dA/dt = 3 - 7A/(500-t)

except after i did solved the d.e. i kept getting more salt than i started with which makes no sense...

bdh2991 said:

## Homework Statement

a larg tank is filled with 500 gals of water with 400lbs of salt. pure water is pumped into the tank at a rate of 3gal/min. the well mixed solution is pumped out at a rate of 7 gal/min.

I need help finding the Differential

## The Attempt at a Solution

I tried coming up with the D.E. and this is what i got dA/dt = 3 - 7A/(500-t)

except after i did solved the d.e. i kept getting more salt than i started with which makes no sense...

Your denominator is wrong. How is rate out defined?

fauboca said:
Your denominator is wrong. How is rate out defined?

not sure lol...i went off of a similar problem that was gaining instead of losing...

i know that it is supposed to be the flow rate out multiplied by the concentration out but i don't understand how to get the concentration of salt exiting the tank...

bdh2991 said:
not sure lol...i went off of a similar problem that was gaining instead of losing...

i know that it is supposed to be the flow rate out multiplied by the concentration out but i don't understand how to get the concentration of salt exiting the tank...

$$R_{out} = -t(7gal/min - 3gal/min) = -4t$$

Where you have 500-t. I didn't check over the rest of you equation but that was what was immediate obvious.

ok so instead i should have A/(-4t) rather than A/(500-t)?

bdh2991 said:
ok so instead i should have A/(-4t) rather than A/(500-t)?

No, why would you disregard the amount of solution?

i guess I'm misunderstanding what you are saying...it would help more if you just showed me what the equation should look like

bdh2991 said:
i guess I'm misunderstanding what you are saying...it would help more if you just showed me what the equation should look like

I could have phrased it better.

When you are dealing with mixtures, you denominators is

$$\frac{A}{sol+(R_{in}-R_{out})t}$$

When inflow and out flow are the same, we just have A/sol in gallons.

Your R_{in}=3 and out=7 so (3-7)t = -4t.

So what is the denominator?

A/500-(-4t)

so A/500+4t?

it can't be that because i just tried it and it didn't come out right

fauboca said:
I could have phrased it better.

When you are dealing with mixtures, you denominators is

$$\frac{A}{sol+(R_{in}-R_{out})t}$$

When inflow and out flow are the same, we just have A/sol in gallons.

Your R_{in}=3 and out=7 so (3-7)t = -4t.

So what is the denominator?

500+(-4t) = 500-4t

ok i made some dumb mistakes but i finally got what i needed thanks!

## 1. What is a mixing tank in the context of differential equations?

A mixing tank is a physical system that involves the combination of two or more substances, such as liquids or gases, in a closed container. In the context of differential equations, it is used to model the behavior of the substances within the tank over time.

## 2. How do you represent a mixing tank using differential equations?

A mixing tank can be represented using a system of ordinary differential equations, where the variables represent the concentrations of the substances in the tank and the parameters represent the rates at which the substances enter or leave the tank.

## 3. What are the applications of solving a differential equations problem involving a mixing tank?

The solutions to differential equations problems involving mixing tanks can be applied in various fields, such as chemical engineering, environmental science, and pharmaceutical research. They can help predict the behavior of substances in real-life systems, optimize processes, and develop new products.

## 4. How do you solve a differential equations problem involving a mixing tank?

The solution to a differential equations problem involving a mixing tank usually involves setting up the system of equations, solving it using appropriate mathematical techniques, and interpreting the results in the context of the problem. Computer simulations and numerical methods may also be used for more complex systems.

## 5. What are some challenges in solving a differential equations problem involving a mixing tank?

Some challenges in solving these types of problems include accurately modeling the behavior of the substances in the tank, choosing appropriate initial conditions and parameter values, and dealing with non-linear or time-dependent systems. It may also be challenging to interpret the results and make connections to real-world applications.

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