Problema Solution
suppose you are cutting 10 identical boards to be 7 feet. The procedure is to measure and cut the first board and then use the first board to measure and cut the second board, and so on till you cut all 10. What are the possible longest and shortest lengths of the 10th board if each time you cut a board there is a maximum error of +/- 0.7%?
Answer provided by our tutors
0.7% = 0.007 as a decimal
there are several ways to look at this problem... for example, am I measure 3 feet of wood to cut? or 7 feet of wood to retain? because 0.7% of 3 is going to be smaller than 0.7% of 7
we choose to look at the minimum error as a 0.7% cut on removal of 3 feet and the maximum error as a 0.7% cut on retaining 7 feet
the accumulation of error can be treated like accumulation of interest, so we use this formula:
on a 7-foot cut:
total error = 7(0.007/(1-(1+0.007)^-10) = 0.727
on a 3-foot cut:
total error = 3(0.007/(1-(1+0.007)^-10) = 0.312
so the length closest to seven foot on the 10th board would be 7 +/- 0.312 feet
and the greatest length with the greatest amount of error would 7+/- 0.727 feet
the shortest length would be 7-0.727 = 6.273 feet
the greatest length would be 7+0.727 = 7.727 feet