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(一)問題的提出电焊工人要焊接一条相交成直角,总长为2l,直径为d的磬形鉄管(如图1a)。現在有两块同样大小的鉄皮,它們的长都是l,寬都是2c=πd。为了便于操作,一般都在焊接前,把二块鉄皮割成如图1b所示的形状。如果割得很准确,那末把两块鉄皮焊成两根直管后,割过的那两端凑攏来很吻合,而且两管刚好成直角,很易焊接。但普通是根据經驗剪割,常常需要在焊接成直管后,再稍加修改才能吻合。这样一方面会損失一些材料,另一方面出会多化一些工夫。因此最好能求出这曲綫的准确形状。 (二)这曲綫函数的解析表示我們取这两块鉄皮中的一块來看。設这曲綫已准确作出,再在这曲綫
(I) Proposal of the problem Welders are required to weld a 磬-shaped 鉄 tube that intersects at a right angle and has a total length of 2 l and a diameter of d (Fig. 1a). There are now two similarly sized furs, all of which have a length of l and a width of 2c=πd. For ease of operation, two pieces of suede are generally cut into the shape shown in Figure 1b before welding. If the cutting is very accurate, then after welding the two pieces of skin into two straight tubes, the cut ends are closely matched to each other, and the two tubes are just at a right angle and are easily welded. However, ordinary cutting is based on experience, and often requires a slight modification after welding into a straight tube. On the one hand, some materials will be lost. On the other hand, there will be more efforts. It is therefore best to find the exact shape of this curve. (B) The analysis of this curve function means we take a look at one of the two pieces of skin. Let this curve be accurately made, and then in this curve