1. Mr. X comes home after a long day of teaching at a university and decides to take a relaxing bath. He fills his bathtub, which initially holds 100 liters of pure water. As he settles in, he notices that the water flowing into the tub isn't just plain water \textendash{} it's mixed with a bit of shampoo that somehow got into the water supply, and the concentration of shampoo changes over time.
The water flows into the bathtub at a steady rate of 5 liters per minute, and the same amount drains out, keeping the water level constant. However, the concentration of shampoo in the incoming water isn't constant. Instead, it varies according to the equation:
$C(t) = \textcircled{\text{\ }}+ \sin(3t)$, where $t$ denotes the time in minutes, $C(t)$ is the concentration of shampoo in grams per liter, and $\textcircled{\text{\ }}$ is the last digit of your student number; e.g., if your student number is B2XXX.012345, then replace $\textcircled{\text{\ }}$ by 5 (i.e., take $C(t) = 5 + \sin(3t)$).
Now, Mr. X is curious: how much shampoo will be in his bathtub as time passes?
Your tasks:
1. Help Mr. X by writing down a differential equation that models the amount of shampoo in the bathtub over time, $S(t)$, in grams.
2. Solve the equation to find how the amount of shampoo changes over time.
