- 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.
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).
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.