A designer proposed using a paraboloid S: x² + y? = z as the basis of a container design. However, after some discussion, the designer was advised to revert to a matching cone-based design S, with...

Extracted text: A designer proposed using a paraboloid S: x² + y? = z as the basis of a container design. However, after some discussion, the designer was advised to revert to a matching cone-based design S, with certain apex angle a, which was said to be able to achieve material cost saving due to less surface area. Figure 1 gives a visual comparison of the two surfaces. 12 14 radius 0.8 0.6 S2: Cone 04 apex langle a S1: Paraboloid x + y² = z 02 -12 --0.8 -0.6 -0.4 -0.2 04 0.6 08 12 Figure 1: Visual representations* of the paraboloid surface and its matching cone surface *Note: the graphs are for illustrative purpose only; the intersection between S, and S, depends on a and does not always occur at z = 1


Extracted text: With your knowledge in surface integral**, determine the surface area of the cone (S2) and the surface area of the paraboloid (S,), and find out the surface area saving. Be sure to refer Table 1 for the value of the apex angle a. Digit a (*) 4 40 **Note: it is recommended to use suitable parameterization for ease of evaluating the two surface integrals: e.g. a possible approach is to view the cone as a circular revolution of a slanted line around the central-axis, so that for every elevation value z, the x and y are related by the basic circular equation x² + y² = (radius)² (see Figure 1). This revolution approach is applicable to the paraboloid as well. END