2017 HiMCM B題特等獎學生論文下載8121
2017 HiMCM B題特等獎學生論文下載8121
下載方式見文末
論文摘要如下：
20th? ?Annual? ?High? ?School? ?Mathematical? ?Contest? ?in? ?Modeling? ?(HiMCM)
?Summary? ?Sheet Team? ?Control? ?Number:? ?8121
Problem? ?Chosen:? ?B:? ?Ski? ?Slope
Summary:
Wasatch? ?Peaks? ?Ranch? ?is? ?a? ?prime? ?location? ?for? ?a? ?ski? ?resort.? ?? ?Given? ?the? ?magnitude? ?of? ?the property? ?and? ?the? ?randomness? ?of? ?nature,? ?it? ?is? ?impossible? ?to? ?find? ?a? ?perfect? ?ski? ?resort? ?layout.? ?? ?Our complete? ?ski? ?slope? ?design? ?is? ?of? ?Olympic? ?caliber? ?and? ?will? ?compete? ?as? ?one? ?of? ?the? ?top? ?ski? ?resorts? ?in all? ?of? ?North? ?America.
Our? ?model? ?begins? ?with? ?extensive? ?path? ?generation? ?which? ?entails? ?hundreds? ?of? ?trails? ?being created,? ?covering? ?nearly? ?60? ?km.? ?A? ?subset? ?of? ?these? ?trails? ?were? ?chosen? ?based? ?on? ?their? ?Trail? ?Quality Index? ?(TQI),? ?a? ?single? ?value? ?which? ?determines? ?the? ?appropriateness? ?of? ?trails? ?for? ?skiers? ?given? ?their skill? ?level,? ?sharpness? ?of? ?curves,? ?width? ?of? ?narrowest? ?point,? ?roughness,? ?steepness,? ?flatness, variability? ?in? ?directional? ?derivatives,? ?and? ?points? ?of? ?kinetic? ?energy? ?loss.? ?Using? ?the? ?data? ?for? ?the? ?40 km? ?subset? ?of? ?trails,? ?we? ?simulated? ?120? ?km? ?of? ?trails? ?using? ?normal? ?distributions? ?for? ?each? ?of? ?the aforementioned? ?values.? ?Our? ?final? ?design? ?boasts? ?more? ?than? ?162? ?km? ?of? ?slopes? ?with? ?38? ?beginner trails? ?(20.4%),? ?74? ?intermediate? ?trails? ?(39.8%),? ?and? ?74? ?expert? ?trails? ?(39.8%).? ?Our? ?beginner? ?and intermediate? ?trails? ?have? ?an? ?mean? ?TQI? ?of? ?3.7? ?compared? ?to? ?a? ?TQI? ?of? ?2.4? ?for? ?Whistler? ?Blackcomb’s premier? ?ski? ?trail.? ?Any? ?TQI? ?values? ?greater? ?than? ?0.5? ?are? ?considered? ?acceptable.
The? ?aggregate? ?of? ?all? ?trails? ?is? ?our? ?Wasatch? ?Peaks? ?Ranch? ?ski? ?resort? ?design.? ?To quantitatively? ?compare? ?our? ?resort? ?with? ?others,? ?we? ?created? ?a? ?Resort? ?Rating? ?system? ?which? ?rates? ?the resorts? ?from? ?0.1? ?to? ?10? ?(10? ?being? ?the? ?best),? ?based? ?on? ?a? ?variety? ?of? ?factors,? ?including? ?ski? ?lifts,? ?trail density,? ?vertical? ?drop,? ?and? ?direction? ?that? ?the? ?slope? ?is? ?facing,? ?which? ?is? ?critical? ?for? ?snow? ?quality. Using? ?this? ?system,? ?Wasatch? ?Peaks? ?Ranch? ?achieved? ?a? ?Resort? ?Rating? ?score? ?of? ?7.44,? ?ranking? ?3rd among? ?16? ?top? ?North? ?American? ?ski? ?resorts.? ?To? ?put? ?this? ?into? ?more? ?context,? ?Park? ?City? ?Mountain, known? ?as? ?one? ?of? ?the? ?best? ?ski? ?resorts,? ?and? ?previous? ?Winter? ?Olympic? ?site,? ?received? ?a? ?score? ?of? ?8.67 using? ?our? ?rating? ?and? ?was? ?ranked? ?#1.? ?Sugarloaf? ?Mountain,? ?ranked? ?extremely? ?poorly? ?on ZRankings.com,? ?received? ?a? ?4.74.? ?The? ?system? ?is? ?proven? ?reliable? ?through? ?its? ?strong? ?correlation with? ?existing? ?ranking? ?systems? ?and? ?its? ?vulnerability? ?to? ?sensitivity? ?analysis.
Our? ?rating? ?systems? ?are? ?based? ?on? ?weighted? ?composite? ?scores? ?in? ?the? ?various? ?factors? ?we incorporated,? ?so? ?that? ?it? ?truly? ?reflects? ?what? ?skiers? ?are? ?looking? ?for.? ?It? ?also? ?incorporates? ?logistic models? ?to? ?make? ?sure? ?that? ?ratings? ?below? ?the? ?minimum? ?and? ?ratings? ?above? ?the? ?maximum? ?cannot happen.
To? ?measure? ?Olympic? ?compatibility? ?and? ?readiness,? ?we? ?used? ?a? ?similar? ?system,? ?which resulted? ?in? ?our? ?Olympic? ?readiness? ?score? ?of? ?7.8,? ?well? ?within? ?the? ?range? ?of? ?previous? ?Olympic? ?sites: Squaw? ?Valley’s? ?7.0? ?to? ?Park? ?City? ?Mountain’s? ?8.3.
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