用半径为R的圆铁皮剪去一个圆心角为a的扇形,制成一个圆锥形容器,扇形的圆心角多大时,容器的容积最大?

361

设圆锥底面半径r 圆锥高h
r=a*r/2pai
h^2=R^2-r^2=...=R/2pai*更号下(4pai^2-a^2)
v=pai*r^2*h/3=[a^2*R^3更号下(4pai^2-a^2)]/24pai^2
v^2=(1/2)*(a^3/24pai^2)*a^2*a^2(8pai-2a^2)
<=(1/2)*(a^3/24pai^2)(8pai^2/3)^3
=4a^3*pai^2/81当且仅当a=2(更号下6)*pai/3时成立

所以 a=2更号下6/3pai