Differential equations-system of equations, cleaning up the great lakes

In summary, the problem at hand is to determine the time it would take to flush the Great Lakes of pollution. The lakes are divided into five tanks and the given information includes inflow rates of clean water, inflow rates from other tanks, and outflow rates. The equations for each lake were written using the first letter of each lake to represent the amount of pollution at time t. This leads to a system of five equations with five variables. However, the next steps for solving this system are unclear.
  • #1
mwaso
11
0

Homework Statement



The idea of the problem is to find out how long it would take to flush the Great Lakes of pollution. They're set up as a series of five tanks and you are given inflow rates of clean water, inflow rates from the other tanks, and outflow rates. See attached figure.


Homework Equations



dA/dt=ratein-rate out where A is the amount of pollution at time t

The Attempt at a Solution



first I wrote equations for each lake. rather than using A as my variable, I used the first letter of each lake (with n for Ontario) to stand for the amount of pollution in the given lake at time t. this gives...

ds/dt=-15s/2900

dm/dt= -38m/1180

dh/dt=15s/2900 + 38m/1180 - 68h/850

de/dt=68h/850 - 85e/116

dn/dt=85e/116 - 99n/393

rearranging and pulling out the differential operator leads to the following system

(D + 38/1180)[m]=0
(D - 68/850)[h] + 38m/1180 - 15s/2900=0
(D + 15/2900)=0
(D + 85/116)[e] -68h/850=0
(D + 99/393)[n] - 85e/116=0

and there you have it. A system of five equations with five variables. It seems like it should be fairly straightforward to solve from here, but I can't figure out what to do next. I'm stuck!
 

Attachments

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  • #2
http://images.quickblogcast.com/56158-49218/great_lakes.JPG [Broken]

here's the image that shows the great lakes and their flow rates so you don't have to view the attachment
 
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1. What are differential equations?

Differential equations are mathematical equations that involve an unknown function and its derivatives. They are used to describe the relationship between a quantity and its rate of change over time.

2. How are differential equations used in the context of cleaning up the great lakes?

In the context of cleaning up the great lakes, differential equations can be used to model the flow and transport of pollutants, as well as the rate of change of water quality over time. By solving these equations, we can better understand the dynamics of the lake system and make predictions about the effectiveness of different clean-up strategies.

3. What is a system of equations?

A system of equations is a set of two or more equations that contain multiple unknown variables. These equations are solved simultaneously to find the values of the unknown variables that satisfy all of the equations.

4. How are systems of equations used in the context of cleaning up the great lakes?

In the context of cleaning up the great lakes, systems of equations can be used to model the interactions between different variables, such as the concentration of pollutants, water flow rates, and changes in water quality. By solving these equations, we can gain a better understanding of the complex relationships within the lake system and make more informed decisions about clean-up efforts.

5. What are some challenges in using differential equations and systems of equations to clean up the great lakes?

One of the main challenges is accurately representing all of the variables and factors that affect the lake system in the equations. There may also be limitations in data availability and uncertainties in the parameters used in the equations. Additionally, solving these equations can be computationally intensive and require advanced mathematical techniques.

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