Final answer:
To determine the quantity of salt in a tank as it's about to overflow, we must solve a differential equation that accounts for the rates of inflow and outflow of the salt solution, using the given initial conditions. By integrating the equation, we can find the amount of salt at the moment when the tank reaches its full capacity.
Step-by-step explanation:
To find the quantity of salt in a 200 gallon tank as it's about to overflow, we need to understand that this is a mixing problem, often studied in differential equations. The tank starts with 100 gallons of water and 20 pounds of salt. A salt solution with 1/4 pound of salt per gallon is added at a rate of 4 gallons per minute, and the mixture leaves the tank at 2 gallons per minute.
Here are the steps to find the amount of salt at the point of overflow:
- Set up the differential equation that represents the situation. Let's call S(t) the amount of salt at time t, and V(t) the volume of water in the tank at time t.
- Since the solution enters the tank at 4 gal/min and the concentration of salt in the incoming solution is 1/4 lb/gal, the rate of salt coming into the tank is (1/4) * 4 = 1 lb/min.
- The tank is being drained at 2 gal/min. So, the rate of salt leaving the tank is given by (S(t)/V(t)) * 2, because the concentration of salt at any time is S(t)/V(t).
- Setup the equation: dS/dt = 1 - (2 * S(t))/V(t). Note that V(t) = 100 + 2t, since the net rate of water entering the tank is 4 gal/min incoming minus 2 gal/min outgoing, for a net increase of 2 gal/min.
- Substitute V(t) into the differential equation and solve for S(t) using the initial condition S(0) = 20.
- Finally, compute S(t) when V(t) equals 200 gallons, which is the point of overflow.
Through this method, we will find the quantity of salt in the tank just as it's about to overflow. This is a mixing problem solved using concepts from calculus and differential equations.