Question

A tank initially contains 40 gallons of pure water. A solution with 1 gram of salt per gallon of water is added to the tank at 3 gal/min, and the resulting solution dranes out at the same rate. Find the quantity Q(t) of salt in the tank at time t > 0.

Answer 1

The quantity Q(t) of salt in the tank at time t is given by Q(t) = 40 + 3t where t is the time in minutes. The initial quantity of salt is 40 grams.

Step-by-step explanation:

To find the quantity Q(t) of salt in the tank at time t, we need to consider the rate at which salt is added and drained from the tank. The solution is being added at a rate of 3 gal/min, so the amount of salt added per minute is 3 grams since the concentration is 1 gram of salt per gallon of water. At the same time, the solution is draining out at the same rate, so the amount of salt drained per minute is also 3 grams. Therefore, the quantity of salt in the tank at time t would be equal to the initial quantity of salt plus the net amount of salt added or drained over time. In this case, the initial quantity of salt is 40 grams, so the quantity of salt at time t is given by Q(t) = 40 + 3t where t is the time in minutes.