Question

A tank with a capacity of 500 gal originally contains 200 gal of water with 100 lb of salt in solution. Water containing 1 lb of salt per gallon is entering at a rate of 3 gal/min, and the mixture is allowed to flow out of the tank at a rate of 2 gal/min. Find the amount of salt in the tank at any time prior to the instant when the solution begins to overflow. Find the concentration (in pounds per gallon) of salt in the tank when it is on the point of overflowing. Compare this concentration with the theoretical limiting concentration if the tank had infinite capacity.

Answer 1

Let A(t) denote the amount of salt in the tank at time t.

Salt flows in at a rate of

(1 lb/gal) * (3 gal/min) = 3 lb/min

and flows out at a rate of

(A(t)/(200 + t) lb/gal) * (2 gal/min) = 2 A(t)/(500 + t)

(in case you're unsure about the denominator: the tank starts off with 200 gal of solution, and each minute solution flows in at a rate of 3 gal/min and thus the tank gains (3 gal/min) * (1 min) = 3 gal. At the same time, solution flows out at a rate of 2 gal/min and thus the tank loses 2 gal, giving a net change in volume of (3 - 2)*t = t gal)

Then the net rate of salt flow is given by the ODE,

$\dfrac{\mathrm dA(t)}{\mathrm dt}-\dfrac{2A(t)}{200+t}=3$

Multiply both sides by $(200+t)^{-2}$:

$(200+t)^{-1}\dfrac{\mathrm dA(t)}{\mathrm dt}-2(200+t)^{-3}A()=3(200+t)^{-2}$

$\implies\dfrac{\mathrm d}{\mathrm dt}\bigg((200+t)^{-2}A(t)\bigg)=3(200+t)^{-2}$

Integrating both sides and solving for $A(t)$ gives

$(200+t)^{-2}A(t)=-\dfrac3{200+t}+C$

$A(t)=-2(200+t)+C(200+t)^2$

The tank starts off with 100 lb of salt in solution, so $A(0)=100$ and we find

$100=-2(200)+C(200)^2\implies C=\dfrac1{80}$

and so

$A(t)=-2(200+t)+\dfrac{(200+t)^2}{80}=\dfrac{(200+t)(40+t)}{80}$

The tank will begin to overflow once the volume of solution reaches 500 gal; this happens when

$500=200+t\implies t=300$

or 300 minutes or 5 hours after solution starts flowing. At this point, the tank will contain

$A(300)=2125$

or 2125 lb of salt.

Theoretically, the amount of salt in the tank will increase forever, since $A(t)\to\infty$ as $t\to\infty$.