$\"\"$

\

The radius of the sphere is $\"\"$ .

\

Assume,

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The radius of the cylinder is $\"\"$ .

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The height of the cylinder is $\"\"$ .

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Phythagoras theorem : $\"\"$ .

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Figure:

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$\"\"$

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Observe the figure :

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$\"\"$

\

$\"\"$

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The volume of the cylinder is $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Find the maximum volume :

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$\"\"$

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$

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Find the maximum volume of the cylinder is $\"\"$ at $\"\"$ .

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$\"\"$

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$\"\"$

\

$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The maximum volume of the cylinder is $\"\"$ .

\

$\"\"$

\

The maximum volume of the cylinder is $\"\"$ .