
Optimization question.
edit:
Can someone please show the steps to get $\displaystyle
f'(r) = \frac{6\pi r}{5}  \frac{200}{r^{2}}
$
when differentiating f(r) = 0.6pi*r^2 + 0.4pi*r*(500 / pi*r^2)
For a homework question i'm doing it asks for me to find the dimensions that will minimize the cost of producing a can with a volume of 500cm^3, given that the cost for the material of the top of the can is 0.4cents/cm^2, the cost for the material of the bottom of the can is 0.2cents/cm^2, and the cost for the material of the siding of the can is also 0.2cents/cm^2. This is what I tried doing so far:
total cost of material = area of top(0.4) + area of bottom(0.2) + area of siding(0.2)
= 0.4pi*r^2 + 0.2pi*r^2 + 0.2(2pi*r*h)
= 0.6pi*r^2 + 0.4pi*r*h
= 0.6pi*r^2 + 0.4pi*r*(500 / pi*r^2)
I knew to sub that in for h because the volume of a cylinder = pi*r^2*h, therefore 500=pi*r^2*h, and h = 500 / pi*r^2
I'm kind of stuck as to what to do now (Worried) Also did I make any mistakes so far?

You went to all that trouble to create f(r) = Cost of Producing the can as a function of r, and now you don't know what to do with it? Come on! Make the leap!!
What section are you in? I'm guessing "applications of the derivative"?
Find f'(r) and see if it is ever zero (0).
1) Make sure it is a minimum and not a maximum
2) Don't forget to check endpoints, since the derivative doesn't know about those. You have 0 < r < 500/pi at least. It is likely you can narrow the Domain much more than that with a little thought.

Would the derivative be 1.2pi*r + 200pi*r^2  400pi*r^3?

?? Maybe! Did you make any deliberate errors?
1) PLEASE simplify first. There is just no reason to have any r^3 in there.
2) You tell me if it is correct. Part of the exercise is your confidence. Only you will be there for your exam.
3) If it were me, I'd get rid of the decimals. Your life will be MUCH easier. 0.6 = 3/5 and 0.4 = 2/5. Much nicer.
4) I get this: $\displaystyle f'(r) = \frac{6\pi r}{5}  \frac{200}{r^{2}}$. Keep at it until you get that.
Perhaps an odd couple of questions that will help all the volunteers help you more effectively.
A) Why are you in this class? Is it a class? Maybe you are on your own?
B) How are you in this class? Is it a sitdown sort of thing or online, or what?
C) Do you have a text? Which one?
Apology  I may seem a little harsh, but try not to take it personally. If you start to talk funny, or don't seem to have a clue which direction to go, I tend to blame it on your teacher. You don't get to do that, since that would distract you from your studies, but I can. (Evilgrin)
