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Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height *h* and semi vertical angle *α* is one-third that of the cone and the greatest volume of cylinder is `4/27 pih^3` tan^{2}*α*.

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#### Solution

The given right circular cone of fixed height (*h)* and semi-vertical angle (*α**)* can be drawn as:

Here, a cylinder of radius *R* and height *H* is inscribed in the cone.

Then, ∠GAO = *α*, OG =* r*, OA = *h*, OE = *R*, and CE = *H*.

We have,

*r *= *h* tan *α*

Now, since ΔAOG is similar to ΔCEG, we have:

Concept: Maximum and Minimum Values of a Function in a Closed Interval

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