In his pottery class, Roy noticed that when cylindrical container with clay is rotated, the clay forms a parabolic shape. After some experimenting, he notices that the surface of the clay is a paraboloid of revolution generated by rotating the parabola  y = h + ω² x²/(2g) about the y-axis, where ω is the angular velocity and g is the gravitational acceleration. He now wonders how he can determine the angular velocity so that the clay will conform to a given height and width. He would like to use this for a mold.

What is known:

Mold size required is 30 cm high and 40 cm wide as shown in the diagram at the left.

Magnitude of the Mold

What is the angular velocity required so the clay is 30 cm high at the outside edge and 5 cm high at the center? Also, what is the total volume of clay needed?

• Use the point A(20, 30) to help determining the angular velocity.
• This is a cylindrical shell (see theory). Use the coordinate system shown in the diagram.