A large tank is filled with 200 gallons of pure water. Brine containing 1 pound of
salt per gallon is pumped into the tank at the rate of 5 gallon per minute. The
well-mixed solution is pumped out at the same rate.
(a) Find the number A(t) of pounds of salt in the tank at any time t.
Please help! I am getting the wrong solution.
To find the number A(t) of pounds of salt in the tank at any time t, we need to consider the rate at which salt is added and removed from the tank.
Let's break down the problem step by step:
1. Determine the rate at which salt is added to the tank:
We are told that brine containing 1 pound of salt per gallon is pumped into the tank at a rate of 5 gallons per minute. Therefore, the rate at which salt is added to the tank is 5 pounds per minute.
2. Determine the rate at which salt is removed from the tank:
The well-mixed solution is pumped out of the tank at the same rate of 5 gallons per minute. Since the solution is well-mixed, the concentration of salt remains constant throughout the tank. Therefore, the rate at which salt is removed from the tank is also 5 pounds per minute.
3. Determine the initial amount of salt in the tank:
The tank is initially filled with 200 gallons of pure water, which does not contain any salt. Therefore, at t=0, the initial amount of salt in the tank is 0 pounds.
4. Use the information above to set up a differential equation:
The rate of change of the amount of salt in the tank (dA/dt) is equal to the rate at which salt is added minus the rate at which salt is removed. In this case, the differential equation is:
dA/dt = 5 - 5 = 0
5. Solve the differential equation:
Since the differential equation is constant and equal to zero, the solution is simply A(t) = 0. This means that the amount of salt in the tank remains constant at 0 pounds throughout time.
Therefore, the number A(t) of pounds of salt in the tank at any time t is always 0.