Answer:
diameter = height ≈ 6.3 m
Explanation:
You want the dimensions of the cylinder with minimum area that has a volume of 200 m³.
Minimum
The short answer is that a cylinder with minimum surface area has a height equal to its diameter. The volume is ...
V = (π/4)d²h = (π/4)d³
Then the diameter is ...
d = ∛(4V/π) = ∛(800/π) ≈ 6.3 . . . . meters
The height and diameter of the cylinder are about 6.3 meters.
Long answer
The volume relates the height to the diameter:
V = (π/4)d²h
h = (4V)/(πd²)
The surface area equation is ...
SA = (π/2)d² +πdh
SA = (π/2)d² +4V/d . . . . . surface area as a function of diameter
Setting the derivative to zero, we have ...
SA' = 0 = πd -4V/d² . . . . . . . derivative with respect to d
4V = πd³ . . . . . . . . . . . . . . . . multiply by d², add 4V
d = ∛(4V/π) = ∛(4·200/π) ≈ 6.3 . . . . as above
h = 4V/(πd²) = d³/d² = d . . . . . generic solution for minimum area
The height and diameter are about 6.3 m.
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Additional comment
Rounded to the nearest tenth, the radius is 3.2 m.
The usual formulas for volume and area make use of the radius, r = d/2. Substituting d/2 for r in those formulas, we get formulas in terms of the diameter. Here, that is convenient because we know the diameter and height will have the same value.
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