The correct option is C 2√3π
Given :
Slant height of the cone, L=3 m.
Let the radius be 'r' and height be 'h'.
Volume of a right circular cone,
V=13πr2h
⇒V=13π(L2−h2)h [∵L2=r2+h2]⇒V=13π(9−h2) h
Differentiating V w.r.t. h and equating to 0, we get
dVdh=13π(9−3h2)=0⇒h=±√3
As height cannot be negative so, h=√3 m
Again differentiating V w.r.t. h,
d2Vdh2=−2πh<0 at h=√3 m
∴ Maximum volume occurs at h=√3.
Hence, the maximum volume of the cone,
V=13π(9−h2) h =13π(9−(√3)2) √3=2√3π m3