arcsinc( 1y ), for small y

07072018, 06:11 AM
(This post was last modified: 08182019 01:14 PM by Albert Chan.)
Post: #5




RE: arcsinc( 1y ), for small y
I reread the Five Minute challenge posts (highly recommended)
I found a simple formula that is slightly better than e = sqrt(6 m d) / 4 Let z = gain in length of of hypotenuse Using post #30 3/4 rule, z = d/2 * 3/4 = 3/8 d e^2 = (m/2 + z)^2  (m/2)^2 = m z + z^2 e = sqrt((m + z) z) With m=5280, d=1: e = sqrt((5280 + 3/8)(3/8)) = 44.49877105 ~ 44.50 ft 3/4 rule is the upper bound for tiny angle, so z (thus e) is overestimated. Let's try with bigger sector angle, say, d = 100 ft: Old formula, e = sqrt(6 * 5280 * 100) / 4 = 444.9719092 ~ 444.97 ft New formula, e = sqrt((5280 + 300/8)(300/8)) = 446.5492694 ~ 446.55 ft We have a nice bound, correct e is between 444.97 ft to 446.55 ft For correct value of e, with big angle, do arcsinc with corrected k: y = 1  5280/5380 = 100/5380 = 0.01858736 k = 0.15 / (1  13/56 y / (1  0.19068 y / (1  0.21627 y))) = 0.1506523749 x = sqrt(6y) / (1  k y) = 0.3348901066 (about 19 degree) e = (m/2) tan( x/2 ) = 446.2332298 ~ 446.23 ft (correct) 

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Messages In This Thread 
arcsinc( 1y ), for small y  Albert Chan  07052018, 11:43 PM
RE: arcsinc( 1y ), for small y  Albert Chan  07062018, 03:22 PM
RE: arcsinc( 1y ), for small y  Thomas Klemm  07062018, 09:07 PM
RE: arcsinc( 1y ), for small y  Albert Chan  07062018, 11:17 PM
RE: arcsinc( 1y ), for small y  Albert Chan  07072018 06:11 AM
RE: arcsinc( 1y ), for small y  Albert Chan  07072018, 06:04 PM
RE: arcsinc( 1y ), for small y  Albert Chan  07082018, 03:28 AM
RE: arcsinc( 1y ), for small y  Albert Chan  07092018, 01:12 AM
RE: arcsinc( 1y ), for small y  Albert Chan  07122018, 05:26 PM
RE: arcsinc( 1y ), for small y  Albert Chan  08202018, 02:20 PM
RE: arcsinc( 1y ), for small y  Albert Chan  08202018, 03:23 PM
RE: arcsinc( 1y ), for small y  Albert Chan  08252018, 03:51 PM
RE: arcsinc( 1y ), for small y  Albert Chan  10012019, 06:03 PM
RE: arcsinc( 1y ), for small y  Albert Chan  06182020, 11:54 PM

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