Final answer:
The dye concentration is reduced by 1/5 each cycle after replacing 1 L of the mixture with fresh water. By repeating this process three times, the amount of dye remaining in the container at the end of the third cycle is 78.08 mL.
Step-by-step explanation:
The question involves a dilution problem where dye is being added to a container of water and then part of the mixture is replaced with fresh water, and this cycle is repeated. We start with 5 L of water and add 40 mL of dye, then remove 1 L and replace it with fresh water.
After the first cycle, 40 mL of dye is diluted in 5 L of solution. When 1 L is removed, we have 4 L of the original mixture plus 1 L of fresh water. The concentration of the dye in the remaining liquid is now 4/5 of the original. Therefore, after one cycle, the amount of dye is 40 mL × (4/5), which equals 32 mL.
Adding 40 mL more of the dye brings the total back to 72 mL. After the second cycle and removal of 1 L, the dye remaining is 72 mL × (4/5), which gives us 57.6 mL.
Adding another 40 mL to this gives 97.6 mL.
Following the third cycle, 97.6 mL × (4/5) equals 78.08 mL.
Therefore, at the end of the third cycle, 78.08 mL of dye is left in the container.