Mixture problem "A tank contains..."
This is a first order differential equation, but it's taught in calc 2. If I'm in the wrong forum, please move me! :)
It is in verbose detail because I'm having trouble with both problems ( 2 problems, 2 hours... (Headbang) ) and hope that it's a similar issue so I can fix both.
"A tank contains 100 gal of fresh water. A solution containing 1 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min and the mixture is pumped out of the tank at the rate of 3 gal/min. Find the maximum amount of fertilizer in the tank and the time required to reach the maximum."
The book's given answer is "y(27.8)=14.8 lb, t = 27.8 min" verbatim.
So, I start out by writing dy/dt = 1 gal/min * 1 lb/gal - 3 gal/min * Y/(100+(1-3)*t) lb/gal.
I simplified that to dy/dt = 1 lb/min - 3*y/(100-2t) lb/min.
Then I chucked my units of measurement and put it into standard form.
dy/dt + 3Y/(100-2t)=1
I used the integrating factor e^integral(3/(100-2t) = e^(3*ln(100-2t) =
This gave me d((100-2t)^-3*y)=(100-2t)^-3.
I integrated, got (100-2t)^-3*y=1/4 * (100-2t)^-2 + C
I multiplied across by (100-2t)^3 and ended up with y=1/4 * (100-2t) + C.
y(0)=100, which means 100=100/4+C, or C=75.
And now I'm completely cornered. Taking the derivative of what I have gets me -1/2, so I can't find out when the slope is 0 and the tank is full, nor go any further with the problem. Where did I go wrong in it?