calculus

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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 -

    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^2-L^2/3)*sqrt(1/3)L
    = 2*pi*L^3/(9*sqrt3)

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