Question

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?

Answer 1

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)