Answer Happy

On June 30th, the year 1276, around 4:00 pm, a system design engineer was making an exquisite metal vase, the secret aesthetic ratios are released in this problem (see the figure). At the end of the process, an electroless nickel immersion gold plating (ENIG) will be applied (he was ahead of his time). The engineer has only 3 gold nuggets in his savings account, and it is estimated that a single gold nugget translates into 80 cm² of plating. Aiming to produce a completely plated vase, he also aimed to maximize the capacity of the vase. Which values for w and I should be used for that purpose? A B D W 2w 2 w Keep in mind the following points: (1) You are advised to consider a coordinate system, with one of the axis being the main vertical axis, and the other being the first horizontal dotted line. (2) Only the outer surface is to be plated, and excluding the bottom. (3) In the figure, the design is divided into stages A, B, C, and D. The corresponding curves are different, but they all meet smoothly. (4) (a) In stage A, there is simply a cylinder of radius w and length L. (b) In stage B, the curve rotated is quadratic, with its peak exactly at the end of stage A. Thus, you may think of this as the function y = x²/2w+w. (c) In stage C, this is a circular curve of radius a and center c (to be determined), say the function is y = √a²(x-c)2. The most important thing here is that the circle curve attaches smoothly with the curve in stage B, this should be enough to determine a and c in terms of w. 1 (d) Finally, in D we just have part of a cone, so the curve is a line. This line again meets the circle part of C smoothly, subject to the prescribed dimensions. Keep in mind that the distance from the center c (of the circular part) to the bottom is 3L/8.