A large tank is partially filled with 100 gallons of fluid in which 20 pounds of salt is dissolved. Brine containing
1
2
pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 45 minutes. (Round your answer to two decimal places.)
To find the number of pounds of salt in the tank after 45 minutes, we need to calculate the amount of salt added to the tank and the amount of salt removed from the tank over that time.
First, let's find the amount of salt added to the tank. The rate at which brine containing 1/2 pound of salt per gallon is pumped into the tank is 6 gallons per minute. Therefore, in 45 minutes, the amount of brine pumped into the tank is:
6 gallons/min * 45 minutes = 270 gallons
The total amount of salt added to the tank is:
270 gallons * 1/2 pound/gallon = 135 pounds
Next, let's find the amount of salt removed from the tank. The rate at which the well-mixed solution is pumped out of the tank is 4 gallons per minute. Therefore, in 45 minutes, the amount of solution pumped out of the tank is:
4 gallons/min * 45 minutes = 180 gallons
Since the solution is well-mixed, the concentration of salt remains constant in the tank. The initial concentration of salt in the tank is 20 pounds/100 gallons, and since 135 pounds of salt has been added to the tank, the total volume of the solution in the tank is:
100 gallons + 270 gallons - 180 gallons = 190 gallons
Finally, let's calculate the amount of salt remaining in the tank after 45 minutes. The total amount of salt in the tank is:
20 pounds + 135 pounds - (1/2 pound/gallon * 180 gallons) = 20 pounds + 135 pounds - 90 pounds = 65 pounds
Therefore, after 45 minutes, there will be 65 pounds of salt in the tank.