Here's a thread for DTBK.

I have a series of calculus problems for my homework involving the optimization of a can's shape. Given that these cans are average cylinders, but the bases are cut from square sheets of metal with dimensions 2r, I have to show that the optimal ratio of height and radius for minimal metal used with given volume V is h/r = 8/pi. I tried using implicit differentiation on V(h,r) and M(h,r) (M being metal used), then setting M' and V' to zero, simplifying, putting them together in one equation, then simplifying more. This didn't work. Anyone have ideas on what I should do? I could post my work, if it helps. I'm pretty sure I did my product rules right...

Also, equations that may be relevant:
V = hπr^2
M = h2rπ + 2([2r]^2)

EDIT: Pretty much solved that one, I just got -h/r = 8/π, but negative heights can't exist, so that negative just goes away, right?
Anyway, onto the third problem (the second one was the same thing, but the sheets that the bases were cut from were changed to regular hexagons). Now, I have to factor in the cost of joining metal. The cost of the can is proportional to the following:

4sqrt(3)(r^2) + 2πrh + k(4πr + h)

where k is the reciprocal of the length that can be joined for the cost of one unit area of metal. I have to show that cost is minimized when:

V^(1/3)/k = ([πh/r]^[1/3]) ([2π - h/r] / [πh/r - 4sqrt(3)])

WTF is this shit?

Post has been edited 5 time(s), last time on Jan 4 2010, 3:30 am by FatalException.

You could always cheat.

And this is..?

What do you mean?

I was referring to Louie's suggestion of cheating.

The only difference between this and my calculus class (/what I would be doing anyway) is that I've never met any of you IRL. >.>

f'(IRL) = SEN? Interesting.
Also, can you add me on your AIM.

If:
Then V is a non-zero constant.
So V' is zero for any value of h and r. Maybe this will help you?

Yeah, that's why I went for the implicit differentiation and combining. I got it down to h'rπ + hπ + 4r = h'r + 2h (simplified M' and V' respectively) before I started doing algebra on it, but alas, I must be doin' it wrong.

Excel.

I'm currently going to post my current solution for the problem. Currently, I see the sheet of metal having negligible 'height', which means only surface area counts.
V = hπr^2
S = 2πr^2 + 2πrh
I have to show that the optimal ratio of height and radius for minimal metal used with given volume V is h/r = 8/pi.
So I will hypothesize that h/r = 8/pi is the optimal ratio.
hpi = 8r
V = 8r^3
S = 2pir^2 + 16r^2
dV/dr = S = 24r^2
24r^2 = 2pir^2 + 16r^2
8r^2 = 2pir^2
pi = 4
4h = 8r
h = 2r, and since h=2r is the ratio for minimal usage of surface area within a given volume, it fits out.
Thus, h/r = 8/pi is indeed the optimal ratio.
I currently do not have any other solutions other then this.

I'm sleepy, but try defining the metal used at a fixed volume as a function of a ratio lambda. Differentiate with respect to lambda and try that.. I keep making algebra mistakes so I'm going to sleep.

But I don't know what you are trying to differentiate with respect to when you "differentiate implicity".

For the implicit differentiation, it was with respect to r. Also, ratio lambda?