A mold is constructed by rotating clay in a cylindrical container. The angular velocity and the volume of clay need to be determined.

1. The angular velocity

The curve function is given as

      y = 5 + ω²x² /(2g)

The curve must go through point A, so substitute x = 20 and y = 30 , into the function gives the angular velocity. Notice g = 980.7 cm/s²

      30 = 5 + ω²(20)² /(2(980.7))
      25 = 0.2039ω²
      ω = 11.07 rad/s

Substitute ω into the curve function gives

      y = 5 + 0.0625x²

2. The volume of the clay

The volume is generated by rotating the shadow region on the left around y-axis, which generate a cylindrical shell. The volume of a cylindrical shell is given as

        

Integration the above expression gives

45,550 cm³ clay is needed to build this mold.