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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 -

    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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