Question = "If a snowball melts so that its surface area decreases at the rate of $\displaystyle 1 \frac{cm^2}{min} $, find the rate at which the diamter decreases when the diameter is 10cm."

My math went like this: $\displaystyle \frac{da}{dt} = 1 \frac{cm^2}{min} $; Surface area formula is $\displaystyle S = 4*PI*r^2 $, and I want to the derivative of the rate with respect to the area (I think). $\displaystyle \frac{dr}{da} $.

So I kind of mapped out the chain rule $\displaystyle \frac{da}{dr} = \frac{da}{dt}*\frac{dt}{da}$, and I did this $\displaystyle \frac{da}{dt} = 8PIr \frac{dt}{da} $, and subsituted r = 5cm (the radius) to end up getting $\displaystyle \frac{1}{40pi} $ which was wrong. So, I check out the answer sheet and saw that it was $\displaystyle \frac{1}{20PI} $. Seeing that I was close to the right answer I tweaked my math somewhat like this: $\displaystyle 2 = 8PIr \frac {dt}{da} $, and then solved for $\displaystyle \frac{dt}{da} $.

My issue is that I have no idea what I'm really doing with all these Leibniz notations and I won't have the luxury of an answer sheet on an exam. So, I'm asking if anyone has a chance to simplify and show how the chain rule would work here. More specifically, what notations should I be using, area, time, and the rate? It seems to be a theme in these kinds of problems (that I've been doing anyway) to focus on those 3 measurements.

Thanks.