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 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.

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