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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Part 4 Backmatter 11.Bon voyage! 130 Chapter 11 Bon voyage! The theme of this book is how to understand new fields, whether the field is known gen- erally but is new to you; or the field is new to everyone. In either case, certain ways of thinking promote understanding and long-term learning. This afterword illustrates these ways by using an example that has appeared twice in the book the volume of a pyramid. 11.1 Remember nothing! The volume is proportional to the height, because of the drilling-core argument. So V h . But a dimensionally correct expression for the volume needs two additional lengths. They can come only from b 2 . So V bh 2 . But what is the constant? It turns out to be 1 / 3. 11.2 Connect to other problems Is that 3 in the denominator new information to remember? No! That piece of information also connects to other problems. First, you can derive it by using special cases, which is the subject of Section 8.1 . Second, 3 is also the dimensionality of space. That fact is not a coincidence. Consider the simpler but analogous problem of the area of a triangle. Its area is 1 A = bh . 2 The area has a similar form as the volume of the pyramid: A constant times a factor related to the base times the height. In two dimensions the constant is 1 / 2. So the 1 / 3 is likely to arise from the dimensionality of space. That analysis makes the 3 easy to remember and thereby the whole formula for the volume. But there are two follow-up questions. The first is: Why does the dimensionality of space matter? The special-cases argument explains it because you need pyramids for each di- rection of space (I say no more for the moment until we do the special-cases argument in lecture!). 11 Bon voyage! 131 The second follow-up question is: Does the 3 occur in other problems and for the same reason? A related place is the volume of a sphere 4 3 V = r ....
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