calculus
posted by anu on .
A right triangle of hypotenuse L is rotated about one of its legs to generate a right circular cone
find the largest volume that such a cone could occupy

Let cone height be h, which is one of the two legs. The base radius will be
R = sqrt(L^2  h^2)
Volume = (1/3)pi*R^2*h
= pi*(L^2  h^2)*h/3
Volume is a maximum when dV/dh = 0
dV/dh = pi*L^2  pi*3*h^2 = 0
L^2 = 3 h^2
Max volume = (1/3) pi*(L^2L^2/3)*sqrt(1/3)L
= 2*pi*L^3/(9*sqrt3)