Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan-1 √2.


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)


V =






.



Now, if


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 .


Hence Proved.


25
1