A tank holds 300 gallons of a brine solution, the rate at which brine is pumped into the tank is 3 gal/min with a concentration of salt of 2 lb/gal and the well stirred solution is pumped out at a rate of 2 gal/in. It stands to reason that since brine is accumulating in the tank at the rate of 1 gal/in, any finite tank must eventually overflow. Suppose that the tank has an open top and has a total capacity of 400 gallons and A(0)=50.

(a) When will the tank overflow?

(b) What will be the number of pounds in the tank at the instant it overflows?

(c) Assume that although the tank is overflowing, brine solution continues to be pumped in at a rate of 3 gal/min and the solution continues to be pumped out at a rate of 2 gal/min. Devise a method for determining the number of pounds of salt in the tank at t = 150 minutes.

(d) Determine the number of pounds of salt in the tank as t approaches infinity. Does your answer agree with your intuition?

(e) Use a graphing utility to plot the graph of A(t) on the interval [0,500)

First I set up a DE using the information provided

$\displaystyle DA/dt = 6 - (2A/(300 + t))$

$\displaystyle DA/dt + (2A/(300 + t)) = 6$

Used an integrating factor of $\displaystyle (300 + t)^2$

Multiplied by IF and integrated and cleaned it up to get:

$\displaystyle A= 600 + 2t + C/(300+t)^2$ used initial value A(0)=0 to solve for C

$\displaystyle A= 600 + 2t - (4.95x10^7)/(300+t)^2$

(a) tank overflows when 300+t = 400 so when t=100 minutes

(b) A(100)=490 lbs

(c) This is where I get iffy, I'm pretty sure you can't just use A(150) because you need to take into account that the tank is overflowing, I just don't know how to.

(d) I did the limit as t approaches infinity of A(t) so:

$\displaystyle 600 + 2(inf) - 0 = infinity$

obviously since the container has a finite capacity, it's impossible for the amount of salt to be infinity, the problem is that the equation doesn't account for the container's limited capacity

(e) I plugged my equation A(t) into my calculator and basically just got a straight line going diagonally across the screen

Thanks in advance