feb13a - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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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 .
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of a bill. To find the density, use what you know: Money is paper. Paper is wood or fabric, except for many complex processing stages whose analysis is far beyond the scope of this book. When a process, here papermaking, looks formidable, forget about it and hope that you’ll be okay anyway. More important is to get an estimate; correct the egregiously inaccurate assumptions later (if ever). How dense is wood? Wood barely floats, so its density is roughly that of water, which is ρ 1 g cm 3 . So the density of a $100 bill is roughly 1 g cm 3 . 2 Assorted subproblems 13 value/mass for $100 value mass density volume length width thickness sheets in a ream ream thickness The ream (500 sheets) is roughly 5 cm thick. The only missing leaf value is the density volume of a bill is roughly V 6 cm × 15 cm × 10 2 cm 1 cm 3 . So the mass is m 1 cm 3 × 1 g cm 3 1 g . How simple! Therefore the value per mass of a $100 bill is $100/g. To choose between the bills and gold, compare that value to the value per mass of gold. Unfortunately our figure for gold is in dollars per ounce rather than per gram. Fortunately one ounce is roughly 27 g so $800/oz is roughly $30/g. Moral: Take the $100 bills but leave the $20 bills. 2.5 Random walks Here is a tree including all the leaf values: value/mass for $100 value $100 mass density 1 g cm 3 volume length 6 cm width 15 cm thickness sheets in a ream 500 ream thickness 5 cm Now propagate the leaf values upward. The thickness of a bill is roughly 10 2 cm, so the
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6.055 / Art of approximation 14 The estimates in Section 2.1 and Section 2.3 are surprisingly accurate. The true pit spacing in a CDROM varies from 1 µ m to 3 µ
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