SOLUTION: HI! please help me with this. I am really CLULESS on this question. Pineapple chunks are often packed in Cylindrical cans. If each can holds 500 ml determine a) The dimensions

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Question 117363: HI!
please help me with this. I am really CLULESS on this question.
Pineapple chunks are often packed in Cylindrical cans. If each can holds 500 ml determine
a) The dimensions to one decimal place, of the can with minimum volume
b) The cost to the nearest penny if the aluminum costs $ 0.2/ cm(squared)

Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
Basically you have to find the relation between the radius and height of a right circular cylinder with given volume for minimum surface area.

Cans are hollow cylinders with both ends closed.
So, total surface area is S+=+2%2Api%2Ar%5E2+%2B+2%2Api%2Ar%2Ah.
S+=+2%2Api%2Ar%28r+%2B+h%29
The volume is V+=+pi%2Ar%5E2%2Ah
h+=+V%2F%28pi%2Ar%5E2%29

So, S+=+2%2Api%2Ar%28r+%2B+V%2F%28pi%2Ar%5E2%29%29 [substituting the value of 'h']
S+=+2pi%2Ar%5E2+%2B+2V%2Fr

For minimum surface area,
dS%2Fdr+=+0
4%2Api%2Ar+-+2V%2Fr%5E2+=+0
r+=+root%283%2CV%2F%282%2Api%29%29

So, the height is h+=+V%2F%28pi%2A%28root%283%2CV%2F%282%2Api%29%29%29%5E2%29
h+=+V%2F%28pi%2Aroot%283%2CV%5E2%2F%284%2Api%5E2%29%29%29
h+=+root%283%2C4V%2Fpi%29

a) Substituting V = 500 cc,
r+=+root%283%2C500%2F%282%2Api%29%29+=+4.30 cm
h+=+root%283%2C4%2A500%2Fpi%29+=+8.60 cm

b) S+=+2%2Api%2A4.3%284.3+%2B+8.6%29+=+348.53 sq cm
So, cost of material per can = $(0.2 x 348.53) = $69.7.