1. ## Mixture problem

A tank initially contains 60 gallons of pure water. Brine containing 1 lb of salt per gallon enters the tank at 2 gal/min, and the perfectly mixed solution leaves the tank at 3 gal/min; thus the tank is empty after exactly 1 hour.

A) Find the amount of salt in the tank after t minutes
B) What is the maximum amount of salt ever in the tank

I am confused as to how to set this up and then after I set it up how would I determin B?

Thanks

2. Originally Posted by mandy123
A tank initially contains 60 gallons of pure water. Brine containing 1 lb of salt per gallon enters the tank at 2 gal/min, and the perfectly mixed solution leaves the tank at 3 gal/min; thus the tank is empty after exactly 1 hour.

A) Find the amount of salt in the tank after t minutes
B) What is the maximum amount of salt ever in the tank

I am confused as to how to set this up and then after I set it up how would I determin B?

Thanks
Here (a). Let $A$ be the amount of salt at time t. So

$\frac{dA}{dt} = r_i c_i - r_o c_o$

rates in and out and concentrations in and out. For your problem since the tank is emptying, V, the colume is changing so

$V = V_0 + (r_i - r_o) t = 60 - t$

$\frac{dA}{dt} = 2 \cdot 1 - 3 \cdot \frac{A}{60-t},\;\;\;A(0) = 0$ - a ODE to solve.