A sequence of three identical barrels each contain 1 m3 of water. The barrels are connected in series at the base so that the first drains into the second, the second drains into the third, and the third is allowed to drain out. It takes one hour for all three barrels to drain completely.
1. How much water (in m3) is in each barrel after t hours (0 ≤ t ≤ 1)?
2. At what rate (in m3/hr) is the water flowing out of the third barrel? The second? The first?
3. If there is initially 1 kg of salt mixed into the first barrel, find the amount of salt in the first barrel after t hours.
4. Find the amount of salt in the second barrel after t hours.
5. Find the amount of salt in the third barrel after t hours.
6. Find the concentration of salt (in kg/m3) in each barrel after t hours.
7. Which barrel has the lowest concentration of salt? What happens to the concentration as t → 1?
Bonus Assume there are n identical barrels connected in series in this way, with 1 kg of salt initially in the first barrel. If it takes one hour for all of the barrels to drain, show that the concentration of salt in the kth barrel is given by Ck (t) = tk-1.