Question

1) A large tank initially contains 50 gallons of a brine solution with a concentration of 2 pounds of salt per gallon of water. Pure water is pumped into the tank at the rate of 4 gallons per minute. the well-stirred mixture is then pumped out at the same rate. Write down the IVP for the amount A of salt in the tank at time t min.

Answer 1

Differential Equation

Answer 2

Let A(t) be the mass of salt, in pounds, in the tank at time t.
Now we need to determine the input and output rate of the amount of salt in the tank.

The input rate is 0 since it's pure water entering the tank and the output rate is:
4(gal/min)*(A/50)(pnds/gal)=2A/25 (pnds/min).

Now the rate of change will be, dA/dt = -2A/25 (pnds/min) with A(0)=2.

This equation is seperable and can be solved for A to get:
A(t)=2e^(-2t/25) with the concentration of salt being A(t)/50=e^(-2t/25)/50.

Answer 3

Sorry, meant A(t)/50=e^(-2t/25)/25.