Q1... Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan-1(√2).
Q2... . Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin-1(1/3)
106(1) the cone has given slant height let it be equal to l
.. semi-vertical angle = θ.
So, r = lsinθ
and h = lcosθ
volume = (1/3)πr2h = πl3sin2θcosθ/3
it is max. when dV/dθ = 0
==> -sin3θ + 2sinθcos2θ = 0
as sinθ≠0
2cos2θ = sin2θ
==> tan2θ = 2
==> tanθ = √2 [not -√2 bcz semivertical angle less than 45]
1The second one is quite long,i'll tell u the method
find a relation between L and (S,R),and from that find H
substitute in Volume eqn
find R for max volume,u will get it in terms of S
now find L from the earlier relation
compare L and R ,u'll get R/L =1/3
