Given a right circular cone with the line of symmetry along x = 0, and the base along y = 0, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the cone? Maximising the volume of the paraboloid relative to the volume of the right circular cone. In 2-D, the parabola has 2 points of tangency to the triangle, one of each side of the line of symmetry. I have tried using the disk method to find the volume of the cone, and the parabola, both with arbitrary equations such as y = b − a x, and y = c − d x 2 , but I end up with a massive equation for several variables, instead of a simple percentage answer. Any help is appreciated! Thanks in advance.

Question

Given a right circular cone with the line of symmetry along   x  =  0, and the base along   y  =  0, how can I find the maximum volume paraboloid (parabola revolved around the y-axis) inscribed within the cone? Maximising the volume of the paraboloid relative to the volume of the right circular cone. In 2-D, the parabola has 2 points of tangency to the triangle, one of each side of the line of symmetry. I have tried using the disk method to find the volume of the cone, and the parabola, both with arbitrary equations such as   y  =  b  −  a  x, and   y  =  c  −  d      x    2  , but I end up with a massive equation for several variables, instead of a simple percentage answer. Any help is appreciated! Thanks in advance.

Jaylon Richmond · Accepted Answer

Hint:Note that here   a and   b   are given.The tangency condition is      a    2    −  4    d    (  b  −  c  )  =  0If you decide to work with   c  , then  d  =            a      2              4            (      b      −      c      )      The volume of the paraboloids, inscribed in the cone, is  V  (  c  )  =  π      ∫    0    c              c      −      y        d      d  y  =  π      ∫    0    c    (  c  −  y  )              4      (      b      −      c      )              a      2          d  y  =            2      π              a      2          (  b        c    2    −      c    3    )Then solve      V    ′    (  c  )  =  0to find the largest volume for  c  =      2    3      bVerify that      V          m      a      x        =      8    9          V          c      o      n      e

Given a right circular cone with the line of symmetry along x = 0 , and the base along

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