Rates of Change and Differential Equations: Filling and Leaking Water Tank
A 50 gallon tank initially contains 20 gallons of pure water. Salt water solution containing .5lb./gallon is poured in at the rate of 4 gallons/minute. At the same time, a drain at the bottom allows the well mixed salt solution to exit at 2 gallons/minute. How many pounds of salt are in the tank at the precise time the tank is full?
What I've done so far:
- At t = 0, the tank has x(0) = 0 pounds of salt per 20 gallons of pure water
- Salt water flows at 4 gallons/min with 0.5 pounds/gallon --> 2 pounds/min = rate of salt coming in
- Drain takes 2 gallons/min of mixed contents --> 2x(t)/(20+2t) pounds/min is the rate of salt leaving
- Want to first find x(f) such that f is the precisely time at which the tank is full
- Then solve to find the general solution of x(t) and plug t = f in to solve.