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A closed cylindrical can must have volume 54π cubic units. The…
Question
A closed cylindrical can must have volume 54π cubic units. The material for the top and bottom costs twice as much per unit area as the material for the curved side. For the can of minimum total material cost, find the ratio of its height to its base radius (height divided by radius).
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Step-by-step solution
Volume: π r^2 h = 54π so r^2 h = 54, h = 54/r^2.
Let the side cost per unit area be 1, top/bottom be 2.
Top+bottom area is 2π r^2 (cost 2 each) and side area is 2π r h (cost 1).
Total cost C = 2·(2π r^2) + 1·(2π r h) = 4π r^2 + 2π r·(54/r^2) = 4π r^2 + 108π/r.
Then dC/dr = 8π r - 108π/r^2 = 0 gives 8r^3 = 108, r^3 = 27/2.
The height-to-radius ratio is h/r = 54/r^3 = 54/(27/2) = 4.
Final answer4
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