Label the endpoints of the base of the isosceles triangular cross-section of the cone B and C and the remaining vertex (apex) A. Mark the centre of the circle as O and perpendicular bisector of A to meet BC at N.
Radius=AO=BO=CO=2; AN=h; BN=NC=r.
dV/dr=4πr/3+(⅓πr²)(-2r/√(4-r²))+⅔πr√(4-r²)=0 when V is max or min.
So using the common denominator 3√(4-r²):
4π⁻r√(4-r²)-2πr³+2πr(4-r²)=0, just considering the numerator.
4π⁻r√(4-r²)-4πr³+8πr=0. Divide through by 4πr:
√(4-r²)=r²-2. Square both sides:
4-r²=r⁴-4r²+4, -r²=r⁴-4r². Divide through by r²:
-1=r²-4, r²=3, r=√3=1.73 units and h=2+√(4-r²)=2+1=3 units.
V=⅓πr²h=(1/3)π(3)(3)=3π=9.42 cubic units.