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` Tan2α. - Mathematics

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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` tan2α.

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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,

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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NCERT Class 12 Maths
Chapter 6 Application of Derivatives
Exercise 6.6 | Q 18 | Page 243

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