Differential Equation problem setup

[BigFlorida]

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.

 

[SteamKing]

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.

 

[BigFlorida]

@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.

 

[BigFlorida]

I actually figured it out with the help of my DE textbook, the

term should just be s, which actually makes a lot of sense because that is the part I was most unsure about.