A swamp interferes with the direct measurement of a surveyline AB. Hence, an auxiliary point C is established and the distance AC is measured with an accuracy of 1 part in 1000. Then the angles BAC and ABC are found to be 45 and 30 degrees respectively, with a possible error of 2 minutes in each.
Find approximately the greatest possible percentage error in the computed length of AB.
The program uses angle-side-angle formula to solve AB using every combination of minimum, maximum and exact values of the three measurements, and the percent error is calculated. Then the largest of the percent errors is searched by eye.
Table using maximum assumed errors in all three observables, in either direction:
actual
AC BAC ABC AB pct error
999 44.96667 29.96667 1931.2631932614083 -0.03046089569
999 44.96667 30.00000 1929.6186258545840 -0.11558996886
999 44.96667 30.03333 1927.9773694620881 -0.20054765131
999 45.00000 29.96667 1931.5653259384508 -0.01482135749
999 45.00000 30.00000 1929.9198009255585 -0.10000000000
999 45.00000 30.03333 1928.2775888549465 -0.18500715199
999 45.03333 29.96667 1931.8668048505720 0.00078433934
999 45.03333 30.00000 1930.2203227885620 -0.08444384368
999 45.03333 30.03333 1928.5771555956644 -0.16950043644
1000 44.96667 29.96667 1933.1963896510592 0.06960871303
1000 44.96667 30.00000 1931.5501760306147 -0.01560557443
1000 44.96667 30.03333 1929.9072767388270 -0.10064829961
1000 45.00000 29.96667 1933.4988247632141 0.08526390641
1000 45.00000 30.00000 1931.8516525781367 0.00000000000
1000 45.00000 30.03333 1930.2077966515981 -0.08509224424
1000 45.03333 29.96667 1933.8006054560281 0.10088522456
1000 45.03333 30.00000 1932.1524752638259 0.01557172805
1000 45.03333 30.03333 1930.5076632589230 -0.06957000645
1001 44.96667 29.96667 1935.1295860407104 0.16967832174
1001 44.96667 30.00000 1933.4817262066451 0.08437882000
1001 44.96667 30.03333 1931.8371840155655 -0.00074894791
1001 45.00000 29.96667 1935.4323235879770 0.18534917032
1001 45.00000 30.00000 1933.7835042307147 0.10000000000
1001 45.00000 30.03333 1932.1380044482496 0.01482266352
1001 45.03333 29.96667 1935.7344060614839 0.20098610979
1001 45.03333 30.00000 1934.0846277390897 0.11558729978
1001 45.03333 30.03333 1932.4381709221821 0.03036042355
The highest percent error seems to be about 0.2 % or 1 part in 500.
clearvars
for AC=1000
for A=45
for B=30
C=180-A-B;
ABperfect=AC*sind(C)/sind(B);
end
end
end
clc
for AC=999:1001
for A=45-1/30:1/30:45+1/30
for B=30-1/30:1/30:30+1/30
C=180-A-B;
AB=AC*sind(C)/sind(B);
fprintf('%4d %7.5f %7.5f %15.13f %15.11f\n',AC,A,B,AB,100*(AB-ABperfect)/ABperfect )
end
end
end
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Posted by Charlie
on 2023-10-13 12:24:45 |
solution
