A right triangle of hypotenuse L is rotated about one of its legs to generate a right circular cone

Question

find the largest volume that such a cone could occupy

Answer 1

Position the cone so that its legs fall along the x and y axes , so that the height of the cone is x along the x-axis, and the radius of the cone is y , along the y-axis

Volume of a cone = (1/3)πr^2h
= (1/3)π(y^2)(x) , but y^2 = L^2 - x^2

V = (1/3)π (L^2x - x^3)
dV/dx = (1/3)π (L^2 - 3x^2) = 0 for a max of V
3x^2 = L^2
x = L/√3

so V = (1/3)π(L^2 - (L/√3)^3)

I will let you simplify it if necessary.