Differential Equation: Mixing Problem

You have written "s'(t)= 100s(t)/768" but you have not told us what that is supposed to mean! s'(t) is "the amount of carbon monoxide leaving the room at time t". Is that what you intended to write?
  • #1
qw111
3
0

Homework Statement



The air in a small room 12x8x8 ft (768 ft^3 is the volume) is 3% carbon monoxide. Starting at t=0, fresh air containing no carbon monoxide is blown into the root at rate of 100 ft^3/min. If the air in the room flows out at same rate (100 ft^3/min), when will air in room be 0.01%?

Homework Equations



s'(t) = (rate entering)(concentration entering) - (rate exiting)(concentration exiting)

concentration = amount/volume

The Attempt at a Solution



I don't need you to solve the problem for me, but I am having trouble getting started and figuring out what the concentration is.

s'(t) = (rate entering)(concentration entering) - (rate exiting)(concentration exiting)
s'(t) = (100)(?) - (100)(?)

concentration = amount/volume
Since it said at time t=0, fresh air containing *NO* carbon monoxide..., will the concentration entering be 0?
And for the concentration exiting, will it be s(t)/768?

so the s'(t) is: 0 - 100s(t)/768

Is this correct?

I am contemplating about the 3% carbon monoxide, could this be the concentration entering??
 
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  • #2
qw111 said:

Homework Statement



The air in a small room 12x8x8 ft (768 ft^3 is the volume) is 3% carbon monoxide. Starting at t=0, fresh air containing no carbon monoxide is blown into the root at rate of 100 ft^3/min. If the air in the room flows out at same rate (100 ft^3/min), when will air in room be 0.01%?

Homework Equations



s'(t) = (rate entering)(concentration entering) - (rate exiting)(concentration exiting)

concentration = amount/volume

The Attempt at a Solution



I don't need you to solve the problem for me, but I am having trouble getting started and figuring out what the concentration is.

s'(t) = (rate entering)(concentration entering) - (rate exiting)(concentration exiting)
s'(t) = (100)(?) - (100)(?)

concentration = amount/volume
Since it said at time t=0, fresh air containing *NO* carbon monoxide..., will the concentration entering be 0?
Yes. But notice that it does NOT say "at time t= 0"- the problem says "starting at time t= 0". There is fresh air containing NO carbon monoxide entering the room for all positive t.

+And for the concentration exiting, will it be s(t)/768?
Well, that would depend upon what "s(t)" means- and you didn't tell us that!

If s(t) is the amount of carbon monoxide in the room at time t, then the concentration will be s(t)/768 per cubic foot. But s'= ds/dt is amount of carbon monoxide per minute. Since air is flowing out at 100 ft^3/min, it carries with it (s(t)/768 carbon monoxide/ft^3)(100 ft^2/min) carbon monoxide per minute.
so the s'(t) is: 0 - 100s(t)/768

Is this correct?

I am contemplating about the 3% carbon monoxide, could this be the concentration entering??
No, it is the initial value: If s(t) is the amount of carbon monoxide in the room then s(0)= .03(768).

I cannot comment on what
 

1. What is a mixing problem in differential equations?

A mixing problem in differential equations refers to a type of problem that involves finding the rate at which a substance is being mixed or diluted over time. This is typically done by using a differential equation to model the changing concentration of the substance.

2. How do you set up a mixing problem differential equation?

To set up a mixing problem differential equation, you need to first determine the rate at which the substance is being added or removed from the mixture, as well as the initial concentration of the substance. You can then use this information to write a differential equation that relates the rate of change of the concentration to the initial concentration and the rate of addition or removal.

3. What are the key assumptions made in a mixing problem?

The key assumptions made in a mixing problem are that the mixture is well-stirred and that there are no external factors, such as chemical reactions or evaporation, affecting the concentration of the substance. These assumptions allow for the use of a simple differential equation to model the problem.

4. Can a mixing problem be solved analytically?

Yes, a mixing problem can be solved analytically by using techniques such as separation of variables or integrating factors. However, in some cases, the problem may be too complex to be solved analytically and numerical methods may be used instead.

5. How is a mixing problem related to other types of differential equations?

A mixing problem is related to other types of differential equations, such as population growth or radioactive decay, in that they all involve modeling the rate of change of a quantity over time. However, a mixing problem is unique in that it specifically deals with the mixing or dilution of a substance in a fluid over time.

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