Hello, franklen!

They gave us the necessary formulas . . .A grain silo has the shape of a right circular cylinder surmounted by a hemisphere.

If the silo is to have a capacity of ft³, find the radius and height of the silo

that requires the least amount of material to construct.

Hint: The volume of the silo is: . .[1]

. . . and the surface area (including the floor) is: . .[2]

From [1], we have: . .[3]

Substitute into [2]: .

. . which simplifies to: .

Differentiate and equate to zero: .

Muliply by

Then we have: .

From .[3], we have: .

Substitute: .

. . and after a LOT of algebra, we get: .

Fascinating . . . the height is exactly five times the radius.

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