Answer:
radius = height ≈ 8.88 cm
Step-by-step explanation:
If the weight is proportional to the surface area, then we need to find the minimum surface area for the given volume.
The volume of the can is given by ...
V = πr²h . . . . . . for radius r and height h
The surface area includes that of the bottom and the sides:
SA = πr² +2πrh
Solving the first equation for h, we can write the second equation as ...
SA = πr² +2πr(V/(πr²)) = πr² +2V/r
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The minimum area for the volume will be found where the derivative is zero.
d(SA)/dr = 2πr -2V/r² = 0
πr³ = V . . . . . multiply by r²/2 and add V
Then the can radius is ...
r = ∛(V/π)
and the height is ...
h = V/(π(V/π)^(2/3)) = ∛(V/π) = r
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For the given volume, the dimensions are ...
r = h = ∛(2197/π) ≈ 8.8762 cm
The lightest can with that volume has a radius and height of about 8.88 cm.
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<em>Comment on cylinder aspect ratio</em>
You may notice that the lateral area is equal to twice the bottom area for this cylinder. If the can had a top, the minimum weight would be such that the lateral area is twice the total of top and bottom area. In that case, the height is equal to the diameter, not the radius.
