User:ViKey/Grenzwertbüchse

Task

Create a can with a certain volume. The surface should be the lowest possible ensure a minimum amount of material. In this case I used the volume one liter as an example.

Intention

This task could also be useful for companies, because our ideal can would need the lowest possible amount of materials. Big companies can save millions of dollars by using this can.

Formulary

Volume (cylinder): V=h*r²*π

Surface (cylinder): S=2r*π*h+2r²π

Solution

V=h*r²*π=1 
=>h=1/(r²*π)  (1) 

S(h,r)=2r*π*h+2r²*π  (2) 

Insert (1) in (2): 
S(r)=(2r*π/r²*π)+2r²*π=2r²*π+(2/r)  (3)
S'(r)=4r*π-(2/r²)  (4)
0=4r*π-(2/r²)
=>2/r²=4r*π
=>2=4r³*π
=>1/2π=r³
=>r=(1/2π)^1/3=1,16245  (5)

Insert (5) in (1):
h=((2π)^2/3)/π=2^(2/3)*π^(-1/3)=1,08385  (6)
both would be in dm

But what would be the solution, if the volume is not one liter:

V=h*r²*π
=>h=V/(r²*π)=(insert (5)) V/((1/2π)^2/3)=0,74004*V
=>r=(V/h*π)^1/2=(insert (6)) (V/(2π)^2/3)^1/2=0,54193*V^1/2

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  Caption1