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A recent article on CNN details an idea for storing water in cold deserts as a cone of ice.

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The article describes many advantages of a cone shape (easy to form, etc.), and includes this quote:

But a cone has more desirable properties: “It has minimal exposed surface area for the volume of water it contains.”

Is this true? If so, how would you prove that of all surfaces enclosing a fixed volume underneath, a cone has the lowest surface area? I haven’t the faintest idea how to approach a minimization problem like this.

Blue
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templatetypedef
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    Relevant: sphere minimizes surface area to volume ratio https://math.stackexchange.com/questions/1297870/prove-that-the-sphere-is-the-only-closed-surface-in-mathbbr3-that-minimize But perhaps “exposed surface area” is the key phrase here, somehow. – littleO Nov 27 '23 at 04:01
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    It's not at all clear what the constraints are, but for a given volume and a given circular (unexposed) base, surely the minimal exposed surface area would be a spherical cap, not a cone.. After all, this is the shape of a soap bubble resting on a plane. – Robert Israel Nov 27 '23 at 04:31

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