A 30-gallon tank initially contains 15 gallons of brine (water and salt solution) containing 6 pounds of salt. Another brine is pumped into the tank at a rate of 3 gallons per minute with an oscillating concentration of 5 + 2 sin t pounds per gallon, where t is time in minutes since the beginning of the experience. At the same time, the solution in the tank is being well mixed and is leaving the tank at a rate of 2e −3t gallons per minute. Set up an initial value problem for (a) the volume of the solution in the tank at any time as long as it is not empty or full, and (b) the amount of salt in the tank at any time. You must explicitly find the volume of the tank as a function of t, but do not solve to find the amount of salt.

A 30-gallon tank initially contains 15 gallons of brine (water and salt solution) containing 6 pounds of salt. Another brine is pumped into the tank at a rate of 3 gallons per minute with an oscillating concentration of 5 + 2 sin t pounds per gallon, where t is time in minutes since the beginning of the experience. At the same time, the solution in the tank is being well mixed and is leaving the tank at a rate of 2e −3t gallons per minute. Set up an initial value problem for (a) the volume of the solution in the tank at any time as long as it is not empty or full, and (b) the amount of salt in the tank at any time. You must explicitly find the volume of the tank as a function of t, but do not solve to find the amount of salt.