Posted by anu on Friday, January 7, 2011 at 1:38am.
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

calculus  Reiny, Friday, January 7, 2011 at 9:23am
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 xaxis, and the radius of the cone is y , along the yaxis
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.
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