So here is the question.
Jimmy has a hundred liter tank of yellow powder. He carves a hole in the bottom, which empties out the contents of the tank at one liter per minute. At the same time, he attaches a hose to the top that adds one liter of water per minute, so that the tank always contains 100 liters of solution. At any given time, the solution is perfectly mixed. A. What is the function that describes the concentration of yellow powder in solution, where 1 describes the starting value? B. What is the concentration as the time approaches infinity.
I understand that the concentration approaches zero because the powder solution keeps falling out while more and more water dilutes the existing solution further. However I don't understand how to create a function describing this because I don't know how to construct a function based on changing rates.
I suspect that where you say "perfectly dissolved" you mean "perfectly mixed". At least such an assumption is required to solve the problem.
Water is emptying out at one liter per minute. There are $100$ liters, so if the current amount of dissolved powder is $p$, then $p/100$ powder is being removed per minute. This is the rate of change of $p$, i.e. its time derivative, so (measuring time in minutes)
$$ p'=-\frac p{100}\;. $$
This is an ordinary first-order homogeneous linear differential equation with constant coefficients (where "ordinary" means not that you should be familiar with it but that it's not "partial" :-). There's lots of material on those on the Web (e.g. at Wikipedia).
The general solution is $p=c\cdot\mathrm e^{-t/100}$, where $c$ is an arbitrary constant that's determined to be $1$ by the given starting value; so $p=\mathrm e^{-t/100}$.