Show that the semi-vertical angle of the cone fo masimum volume and of given slant height is tan inverse root 2

Dear Student!

Here is the answer to your query.

Let α be the semi vertical angle of a cone of slant height *l*

Then V O = *l* cos α and OA = *l* sin α

Let V be the volume of the cone Then

for maximum or minimum V, we must have

Differentially (1) w. r. t. α we get

The V is maximum when tan α = or α = tan^{–1}

i.e. when the semi vertical angle of the cone is tan^{–1}

Cheers!

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