Question = "If a snowball melts so that its surface area decreases at the rate of , find the rate at which the diamter decreases when the diameter is 10cm."
My math went like this: ; Surface area formula is , and I want to the derivative of the rate with respect to the area (I think). .
So I kind of mapped out the chain rule , and I did this , and subsituted r = 5cm (the radius) to end up getting which was wrong. So, I check out the answer sheet and saw that it was . Seeing that I was close to the right answer I tweaked my math somewhat like this: , and then solved for .
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.