Differential Equation problem setup

In summary, the problem involves finding the salt content of a lake at a given time, given the initial conditions of volume and salt content, and assuming that the salt is uniformly mixed with the water at all times. The equations for the rate of water and salt flow into and out of the lake are used to set up a differential equation for the rate of change of salt content in the lake, which can then be solved to find the salt content at any given time.
  • #1
BigFlorida
41
1

Homework Statement


  1. Water with a small salt content (5 lb in 1000 gal) is flowing into a very salty lake at the rate of 4 · 105 gal per hr. The salty water is flowing out at the rate of 105 gal per hr. If at some time (say t = 0) the volume of the lake is 109 gal, and its salt content is 107 lb, find the salt content at time t. Assume that the salt is mixed uniformly with the water in the lake at all times.

Homework Equations


In my setup I let w be the amount of water present in the lake (in gallons), t be time, and s be the amount of salt present in the lake (in pounds).

for dw/dt I have 4x105 gal/hr coming in and 105 gal/hr leaving which gives me
dw/dt = 3x105 gal/hr.

my initial condition for w is w(0)= 109.

my initial condition for s is s(0) = 107.

getting an equation for ds/dt I have the amount of salt coming into the lake is (5/1000)(4x105) = 2x103 lb/hr. Let's call this equation (I).

The amount of salt leaving the lake is given by:

Currently I have (105)((107+2x103t)/109+3x105t) = (107+ 2x103t)/(104 + 3t). Let's call this equation (II).

Therefore,
ds/dt = (I)-(II).

The Attempt at a Solution


I think the main problem I am running into is my "amount of salt leaving the lake" equation. Something seems very sketchy about it, but I cannot think of any way to modify it. Also, it really bugs me that the problem statement says that "the salt content in the lake is always uniform". This makes me think I should always have a ratio of 10-2 lb/gal, which is giving me a lot of problems with my equation (II). I could easily solve the DE if I could just set it up.

Any help would be greatly appreciated.
 
Last edited:
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  • #2
BigFlorida said:

Homework Statement


  1. Water with a small salt content (5 lb in 1000 gal) is flowing into a very salty lake at the rate of 4 · 105 gal per hr. The salty water is flowing out at the rate of 105 gal per hr. If at some time (say t = 0) the volume of the lake is 109 gal, and its salt content is 107 lb, find the salt content at time t. Assume that the salt is mixed uniformly with the water in the lake at all times.

Homework Equations


In my setup I let w be the amount of water present in the lake (in gallons), t be time, and s be the amount of salt present in the lake (in pounds).

for dw/dt I have 4x105 gal/hr coming in and 105 gal/hr leaving which gives me
dw/dt = 3x105 gal/hr.

my initial condition for w is w(0)= 109.

my initial condition for s is s(0) = 107.

getting an equation for ds/dt I have the amount of salt coming into the lake is (5/1000)(4x105) = 2x103 lb/hr. Let's call this equation (I).

The amount of salt leaving the lake is given by:

Currently I have (105)((107+2x103t)/109+3x105t) = (107+ 2x103t)/(104 + 30t). Let's call this equation (II).

Therefore,
ds/dt = (I)-(II).

The Attempt at a Solution


I think the main problem I am running into is my "amount of salt leaving the lake" equation. Something seems very sketchy about it, but I cannot think of any way to modify it. Also, it really bugs me that the problem statement says that "the salt content in the lake is always uniform". This makes me think I should always have a ratio of 10-2 lb/gal, which is giving me a lot of problems with my equation (II). I could easily solve the DE if I could just set it up.

Any help would be greatly appreciated.
Well, which statement is included in the actual problem statement?

1. "Assume that the salt is mixed uniformly with the water in the lake at all times."
or
2. "the salt content in the lake is always uniform."

These two statements don't necessarily mean the same thing.
 
  • #3
@SteamKing I suppose I may have been misinterpreting that statement. I read it as "Assume that the salt is mixed uniformly with the water in the lake at all times, which implies the salt content in the lake is always uniform." I think I see why A does not imply B in this case now, thank you for pointing that out.
 
  • #4
I actually figured it out with the help of my DE textbook, the
upload_2015-10-18_12-14-7.png
term should just be s, which actually makes a lot of sense because that is the part I was most unsure about.
 
Last edited:
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1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with one or more of its derivatives. It represents a relationship between a quantity and its rate of change, and is commonly used to model change over time.

2. How do I set up a differential equation problem?

To set up a differential equation problem, you first need to identify the variables and their relationships. Then, you can use these variables to construct an equation that represents the rate of change of the dependent variable in terms of the independent variable and other known quantities. You may also need to apply any relevant initial or boundary conditions to fully define the problem.

3. What are initial and boundary conditions in a differential equation problem?

Initial conditions are values of the dependent variable and its derivatives at a specific starting point. They are necessary to fully define the solution to a differential equation problem. Boundary conditions are values of the dependent variable and its derivatives at specific points in the domain of the problem. They are used to impose constraints on the solution and can be used to determine unknown constants in the solution.

4. What methods can I use to solve a differential equation problem?

There are several methods that can be used to solve a differential equation problem, including separation of variables, substitution, integrating factors, and series solutions. The choice of method depends on the type of differential equation and its complexity.

5. How do I check the accuracy of my solution to a differential equation problem?

To check the accuracy of your solution to a differential equation problem, you can substitute the solution into the original equation and verify that it satisfies the equation. You can also compare your solution to known solutions or use numerical methods to approximate the solution and compare it to your analytical solution.

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