Let Ɵ be the semi- vertical angle of the cone.
Let r, h and l be the radius, height and the slant height of the cone respectively.
It is given that slant height is constant.
Now, r = lsinƟ and h = lcosƟ
Then, the volume of the cone (V)
sin3Ɵ = 2sinƟcos2Ɵ
⇒ tan2Ɵ = 2
⇒ tanƟ = √2
Now, when, then tan2Ɵ = 2 or sin2Ɵ = 2cos2Ɵ.
Then, we get
= -4πl3cos3Ɵ < 0 for Ɵ ϵ
Then, by second derivative test, the volume (V) is the maximum when
Therefore, the semi-vertical angle of the cone of the maximum volume and of given slant height is .
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