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Lagrange multipliers

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A hot water tank is a vertical cylinder surmounted by a hemispherical top of the same diameter . The tank is designed to hold 750m^3 of liquid.

Determine using Lagrange multipliers the total height and the diameter of the tank if surface heat loss is to be a minimum.

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Lagrange multipliers are employed.

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Let r be the tank's radius and h the cylinder's height.
The volume ...

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