# Mixture problem

• Feb 4th 2009, 01:06 PM
mandy123
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(Worried)
• Feb 4th 2009, 02:50 PM
Jester
Quote:

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(Worried)

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

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

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

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

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