Team 7416 HiMCM 2017 Page 1 of 24 Summary Olympic ski slopes are difficult to plan, and a ranch in Utah is being evaluated to determine whether or not it should host an event. The measurements and difficulty of the course had to be calculated, and this data had to be compared to other ski resorts in North America. The main slopes were selected to be prominent, easily identifiable, and spread out across the mountain in order to maximize the use of the skiable area. The height of the mountains was measured using the contour lines on the interactive topographical online map and the average slope was calculated using right triangle trigonometry. A zig-zag path was chosen for the main slopes because it allows for the adjustment of difficulty levels. Trails were also added parallel to their corresponding slopes and connected to three main trails which run diagonally across the mountain. If the ranch is going to host an Olympic event, it must be comparable to sixteen competitors in North America. A ranking system was a strong method for assessing the qualifications of each location. The number of skiable acres, total slope distance, and distributions of beginner, intermediate, and difficult slopes were the criteria used to determine the feasibility of using the ranch for the Olympics. Weighting techniques were used according to what was thought to be important, and this final calculation determined the placement of each possible venue in the ranking system. The ranch achieved third place when compared against the other locations in the system, and the best venue had previously hosted the Winter Olympic Games.
Team 7416 HiMCM 2017 Page 2 of 24 Memo The design for the new ski resort at Wasatch Peaks Ranch allows for the Winter Olympics to consider this place as a future site for the Games. The team is well aware of the clients that you are representing, and did their best to meet all the needs of the clients, such as including main slopes of varying lengths, plenty of trails, a total of at least 160 kilometers of slopes, which includes main slopes and trails, and a distribution of slopes at approximately 20% rated beginner, 40% rated intermediate, and 40% rated difficult. When incorporating all of these criteria, the team compared the data collected from the new ski area to given data points of ski resorts in North America. The team proved that the new ski resort at Wasatch Peaks Ranch would be an ideal vacation place, as anyone from beginners to experts may enjoy their time there. Additionally, if the International Olympic Committee were to look for possible host cities in North America for the Winter Games, it would definitely consider Wasatch Peaks Ranch to be the venue for skiing events, as the resort, when compared with its competitors, is one of the best on the continent. The first criteria that the team focused on is that of a total of at least 160 kilometers of slopes (including main slopes and trails). In order to do this, the team found the distances for each of the main slopes and each trail. They found that the total distance for all the trails was 73.89 km, and the total distance for all the slopes was 91.66 km. The total distance was calculated by adding the total trail distance (73.89) and the total slope distance (91.66), which came out to be 165.55 km, thus fitting this criterion. Then, using the percent ratio for each type of skill level (20% of the slopes should be a beginner level, 40% should be an intermediate level, and 40% should be a difficult level), the team used angle values that
Team 7416 HiMCM 2017 Page 3 of 24 corresponded to the turn on that slope to find the percent slopes to fit that ratio. A lower angle led to a smaller percent slope. Thus, the team found that there were 4 beginner slopes, 8 intermediate slopes, and 8 difficult slopes, which fits the ratio criterion given. The team successfully fulfilled the needs of the clients for part 1. In part 2, the team compared and ranked the ski resort they created against the other ski resorts. They did so by specifying what categories they wanted to compare, and weighing them based on how important it was to them. The three categories that the team deemed important were: Skiable Acres, Total Slope Distance, and Difficulty Distribution (how many slopes the resort had for each difficulty level). The team weighed the skiable acres five times as much as they weighed the total slope distance, and weighed the difficulty distribution three times as much as they weighed the total slope distance. They determined that the highest score a resort could receive was 153. The Wasatch Peaks Ranch got a score of 113, third highest in the continent, behind Vail Ski Resort, in Colorado, and Whistler Blackomb, in British Columbia, which was already the venue for a Winter Olympics. The team noticed that the ski village was second in the continent for places that have never hosted the Winter Olympics. Although Salt Lake City has already hosted a Winter Olympics, the skiing competitions occurred at Park City Mountain Ski Resort, which opens up the possibility of Wasatch Peaks Ranch to hold the skiing events if the Winter Games return to Salt Lake City. The team hopes that the work they have done satisfies the wants of the clients and that Wasatch Peaks Ranch eventually hosts the Winter Olympics. Thank you, Team 7416
Team 7416 HiMCM 2017 Page 4 of 24 Table of Contents Summary 1 Memo 2 Problem Restatement 5 Assumptions and Explanations 5 Model Description 7 Results/Analysis 11 Future Research 14 Appendix 15 References 24
Team 7416 HiMCM 2017 Page 5 of 24 Problem Restatement Through the use of a mathematical model, develop Wasatch Peaks Ranch into one of the top North American ski resorts. The Wasatch Peaks Ranch ski area should have main slopes of different lengths, many trails, and at least 160 km of main slopes and trails. Roughly 6-20% of the slopes should be beginner slopes, 25-40% of the slopes should be intermediate slopes, and 40% or more of the slopes should be difficult slopes. Rank the ski area against the following North American ski resorts: Beaver Creek, Big Sky Resort, Breckenridge (Colorado), Breckenridge (British Columbia), Jackson Hole, Killington, Lake Louis, Park City Mountain, Silver Star, Squaw Valley, Steamboat Springs, Sugarloaf Mountain, Sun Peaks, Vail, Whistler Blackomb, and Winter Park Resort. Write a memo describing your proposed design and discuss its ranking against the other ski resorts. Assumptions and Explanations A1) The mountain cannot be reshaped and must retain its original dimensions. The heights and lengths of the mountain cannot be adjusted. - Mountain reshaping, while an excellent way to optimize the slopes and trails to fit the requirements, cannot be done in the development of the new ski area. Even though mountain reshaping can lead to pleasing results mathematically, it is impractical to do because of its high cost.
Team 7416 HiMCM 2017 Page 6 of 24 A2) The heights of the mountain faces decrease at a constant rate from the initial point of the slope to the final point of the slope. - Although the heights of mountain faces are typically rough and do not decrease at a constant rate, the assumption is made for simplicity. A3) The width of the main slopes remains constant throughout the ski resort. - The slopes in the proposed model are of uniform width to simplify process of determining a slope s difficulty level. A4) The trails are parallel to the main slopes. - The trails are parallel to the main slopes because they are supposed to provide a pathway from the bottom of the mountain to the tops of the main slopes. In Figures 1, 2, and 3, the trails are directly to the left of their corresponding main slopes. A5) The width of the trails can vary throughout the mountain. - Because the trails do not have a difficulty level, their width does not need to be uniform. A6) The number of ski lifts is not essential in determining a resort s capability to be a future Olympic venue. - A previous Olympic venue, Ski Resort Jeongseon, only had 4 ski lifts. This low number of lifts leads to the assumption that the number of ski lifts at a resort is not critically important in its qualification as an Olympic venue.
Team 7416 HiMCM 2017 Page 7 of 24 Model Description Figure 1. Final sketch of the model created using MS Paint of the proposed ski resort (yellow lines represent the main slopes, black lines represent hypothetical trails, and purple lines divide the area into 3 sections). The main slopes are chosen by using the provided topographical map (see Figure 4 in Appendix). The initial and final points of each slope are then approximated by using the interactive topographical map on Mirr Ranch group s website. The initial point is located near the top of the mountain and the final point is located near the bottom of the mountain. The length of the line segment between the initial and final points represents the horizontal length of the slope. The horizontal length does not mean the length down the slope, but means the length from the initial point to the final point only in the horizontal direction. The interactive topographical map is used to determine the elevations of each point. The change in height for each main slope is then determined by subtracting the elevation of the final point form the elevation of the initial point. Using the change in height and the horizontal
Team 7416 HiMCM 2017 Page 8 of 24 length in the Pythagorean theorem, the distance travelled down the surface of the mountain is found. The difficulty rating of the main slopes is determined based on their most difficult parts. To identify the most difficult parts of each main slope, the interactive topographic map is used to find the two bold elevation lines (boldness denotes increments of 200 feet) that are closest to each other. The distance between the two elevation lines and the height difference of 200 feet are then used in the Pythagorean theorem to find a number called the small hypotenuse. Afterwards, the percent slope calculated from the two aforementioned lines, which is the percent slope of the steepest part of the main slope, is used to determine the difficulty rating of the main slope. Figure 2. Skier Path Down a Mountain To optimize the distribution of beginner, intermediate, and difficult slopes, zig-zag lines are used as the paths of the skier moving down the mountain (see Figure 2). Using zig-zag lines lowers the percent slope and increases the length of the main slope. The change in elevation and the angle of the turns taken by the skier are used to calculate the length of the zig-zag line.
Team 7416 HiMCM 2017 Page 9 of 24 At the top of a main slope, a skier must rotate a set number of degrees in order to follow the zig-zag path. To rotate enough to move down the main slope in the opposite direction, the skier must rotate twice as much and travel twice as much distance. On the most difficult portion of the main slope, there are a total of five turns. Every turn is twice as long as the first turn (this concept can be seen as 1x + 2x + 2x + 2x + 2x), so there are nine hypotenuses. Hence, the height of all the nine right triangles along the zig-zag path is one-ninth of that portion of the zig-zag path. Because the height and angles of the small right triangles have been found, the hypotenuse can be calculated using the law of sines: s in θ/height = sin 90º/hypotenuse h eight = hypotenuse sin θ h ypotenuse = height/sinθ The hypotenuse is multiplied by nine in order to calculate the length of the aforementioned portion of the zig-zag path. To calculate the length of the entire path, the number of zig-zag sections is found by dividing the change in elevation by 40, multiplying the quotient by 2, and subtracting 1 from the product. These steps are carried out because there are two turn-lengths per section (which is 40 feet high) with the exception of the first turn. Afterwards, the small hypotenuse number is multiplied by the total number of zig-zag lengths to determine the total length of the zig-zag main slope (which is the hypotenuse). Finally, the Pythagorean theorem is used to calculate the length of the horizontal leg. To calculate the percent slope, the height of the main slope is divided by the horizontal leg length.
Team 7416 HiMCM 2017 Page 10 of 24 Figure 3: The large triangle represents the linear path down the mountain. The medium triangle represents the steepest section of the main slope. The small triangle s hypotenuse represents one zig-zag length. The percent slope was found through the use of a program (see Appendix). The program receives the small triangle s hypotenuse and the angle as input. Then, it converts the angle from degrees to radians because the sin() command used in the program only functions with radian values. Afterwards, the program calculates the height of the small triangle. The percent slope of the main slope is calculated. The program is used with different angles for each main slope to determine the difficulty level of that main slope. The trails are planned out after the main slopes. The trails run parallel to their corresponding main slopes (a trail is to the left of its corresponding main slope). Three connecting trails united all of the main slopes in their corresponding third of the map (see Figure 1). The trails retain the initial and final elevations of their corresponding main slopes because they begin and end at approximately the same location as the main slopes. To determine the distance covered by the trail, the same process used to determine the
Team 7416 HiMCM 2017 Page 11 of 24 hypotenuse of a main slope is used for each trail. The hypotenuse is the distance covered by the trail. Results/Analysis After all the lengths of the main slopes are measured, the newly acquired measurements and the given data are used to rank Wasatch Peaks Ranch against other ski resorts in North America. The resorts are ranked according to the number of skiable acres (see Figure 4), total slope distance (see Figure 5), and the distributions of beginner, intermediate, and difficult slopes (see Figure 6). The resorts are given a score between one and 17 for each category because 17 resorts are being compared. Figure 4. This bar graph shows the amount of skiable area for North American ski resorts
Team 7416 HiMCM 2017 Page 12 of 24 Figure 5. This bar graph shows the total slope distance for North American ski resorts The resort with the highest number of skiable acres earns a score of 17 and the resort with the least number of acres earns a score of 1. The same procedure applies to the total slope distance. Figure 6. This bar graph shows the slope difficulty ratios of North American ski resorts
Team 7416 HiMCM 2017 Page 13 of 24 Because the ideal ratio of beginner slopes to intermediate slopes to difficult slopes is 1:2:2, the amount of variance from each part of the ratio has to be calculated for each ski resort. In order to find the percent off the correct distribution ratio, the sum of absolute values (net change) is used in order to find the percent off the correct distribution ratio, as modeled below: p ercent of f = 20 b eginning % + 40 i ntermediate % + 40 dif f icult % The results of this calculation is shown in Figure 7 below: Figure 7. This bar graph shows the distribution of difficulty in North American ski resorts The resorts are each assigned a score based on the variance from the ideal ratio; the resort with the smallest difference is given a score of 17, and the resort with the greatest distance earns a score of 1. The weights are assigned to each criteria based on how valuable the group thinks each standard is. The distribution ratio is found to be three times as important as the total amount of slopes, and the number of acres is worth five times as much as the total length of
Team 7416 HiMCM 2017 Page 14 of 24 slopes. Therefore, the following equation is used to calculate the resort scores (which can fall between 3 and 153): r esort score = 5(Acre Rank) + Slope T otal + 3(Dif f iculty Distribution) Future Research The main part of the procedure which should be improved is the calculation of the slope of the trail. Calculus methods and graph fitting can be used to avoid the assumption that the slope between two points on the mountain was constant. The width of the slopes, the sharpness of the turns, and roughness of the terrain can be added to the model. Money and ski lifts can be considered as well because financial problems often plague Olympic host cities.
Team 7416 HiMCM 2017 Page 15 of 24 Appendix Figure 8. Basic image of the topographical map Table 1. This table shows the large triangle data for slopes in feet Slope Name High Elevation (ft) Low Elevation (ft) Height (ft) Length (ft) Hypotenuse (ft) A 5640 5240 400 7397 7408 B 5680 5400 280 7667 7672 C 6520 5520 1000 10486 10534 D 8120 5560 2560 14240 14468 E 9200 5560 3640 15550 15970
Team 7416 HiMCM 2017 Page 16 of 24 F 9200 5200 4000 19436 19843 G 7860 5200 2660 15296 15526 H 7400 5720 1680 13834 13935 I 7800 5680 2120 12609 12786 J 7600 6240 1360 9050 9152 K 8280 6720 1560 6595 6777 L 8800 6800 2000 7946 8194 M 8080 6640 1440 6600 6755 N 8800 6800 2000 9783 9986 O 9600 7200 2400 9879 10166 P 8400 6800 1600 6653 6842 Q 8200 7000 1200 5021 5163 R 8000 7200 800 6336 6386 S 7120 5240 1880 6373 6644 T 7120 5800 1320 6437 6571 Table 2. This table shows the large triangle data for slopes in kilometers Slope Name Height (km) Length (km) Hypotenuse (km) Total Distance of Slopes (km) A 0.12 2.25 2.26 2.95 B 0.09 2.34 2.34 9.03 C 0.30 3.20 3.21 4.19 D 0.78 4.34 4.41 4.69 E 1.11 4.74 4.87 5.18 F 1.22 5.92 6.05 7.38 G 0.81 4.66 4.73 7.36 H 0.51 4.22 4.25 5.19 I 0.65 3.84 3.90 6.06 J 0.41 2.76 2.79 3.94
Team 7416 HiMCM 2017 Page 17 of 24 K 0.48 2.01 2.07 2.70 L 0.61 2.42 2.50 3.26 M 0.44 2.01 2.06 3.59 N 0.61 2.98 3.04 5.31 O 0.73 3.01 3.10 5.03 P 0.49 2.03 2.09 2.41 Q 0.37 1.53 1.57 2.05 R 0.24 1.93 1.95 2.54 S 0.57 1.94 2.03 4.05 T 0.40 1.96 2.00 4.74 Table 3. This table shows the small triangle data for slopes Slope Name Zigzags Small x (km) Small y (km) Small Hypotenuse (km) A 19 559.68 200.00 594.34 Difficulty Angle Measure( ) Slope Percent (%) intermediat e 50.00 26.70 B 13 739.20 200.00 765.78 beginner 15.00 6.77 C 49 480.48 200.00 520.44 intermediat e 50.00 30.83 D 127 401.28 200.00 448.36 difficult 70.00 46.21 E 181 385.44 200.00 434.24 difficult 70.00 48.04 F 199 348.48 200.00 401.79 difficult 55.00 44.76 G 132 454.08 200.00 496.17 intermediat e 40.00 26.84 H 83 380.16 200.00 429.56 difficult 55.00 41.21 I 105 457.26 200.00 499.09 intermediat e 40.00 26.66 J 67 644.16 200.00 674.49 beginner 45.00 21.46 K 77 322.08 200.00 379.12 difficult 50.00 44.20
Team 7416 HiMCM 2017 Page 18 of 24 L 99 451.18 200.00 493.52 intermediat e 50.00 32.62 M 71 522.72 200.00 559.68 beginner 35.00 20.93 N 99 517.44 200.00 554.75 beginner 35.00 21.13 O 119 190.08 200.00 275.92 difficult 38.00 48.85 P 79 417.12 200.00 462.59 difficult 60.00 40.34 Q 59 454.08 200.00 496.17 R 39 506.88 200.00 544.91 intermediat e 50.00 32.48 intermediat e 50.00 29.29 S 93 221.76 200.00 298.63 difficult 30.00 50.45 T 65 290.40 200.00 352.61 intermediat e 25.00 30.47 Table 4. This table shows the distance covered by each difficulty Difficulty Distance (km) beginner 21.88 intermediate 33.16 difficult 36.63 Table 5. This table shows the measurements for the trails in feet Trail Name High Elevation (ft) Low Elevation (ft) Height (ft) Length (ft) Hypotenuse (ft) A 5640 5240 400 7397 7408 B 5680 5400 280 7667 7672
Team 7416 HiMCM 2017 Page 19 of 24 C 6520 5520 1000 10486 10534 D 8120 5560 2560 14240 14468 E 9200 5560 3640 15550 15970 F 9200 5200 4000 19436 19843 G 7860 5200 2660 15296 15526 H 7400 5720 1680 13834 13935 I 7800 5680 2120 12609 12786 J 7600 6240 1360 9050 9152 K 8280 6720 1560 6595 6777 L 8800 6800 2000 7946 8194 M 8080 6640 1440 6600 6755 N 8800 6800 2000 9783 9986 O 9600 7200 2400 9879 10166 P 8400 6800 1600 6653 6842 Q 8200 7000 1200 5021 5163 R 8000 7200 800 6336 6386 S 7120 5240 1880 6373 6644 T 7120 5800 1320 6437 6571 U (Connector 1) 6920 5200 1720 15555 15650 V (Connector 2) 7400 5600 1800 15734 15837 W (Connector 3) 7920 6480 1440 10043 10145
Team 7416 HiMCM 2017 Page 20 of 24 Table 6. This table shows the measurements for the trails in kilometers Height (km) Length (km) Hypotenuse (km) 0.12 2.25 2.26 0.09 2.34 2.34 0.30 3.20 3.21 0.78 4.34 4.41 1.11 4.74 4.87 1.22 5.92 6.05 0.81 4.66 4.73 0.51 4.22 4.25 0.65 3.84 3.90 0.41 2.76 2.79 0.48 2.01 2.07 0.61 2.42 2.50 0.44 2.01 2.06 0.61 2.98 3.04 0.73 3.01 3.10 0.49 2.03 2.09 0.37 1.53 1.57 0.24 1.93 1.95 0.57 1.94 2.03 0.40 1.96 2.00 0.52 4.74 4.77 0.55 4.80 4.83 0.44 3.06 3.09
Team 7416 HiMCM 2017 Page 21 of 24 Table 7. This table shows the total distance covered by the trails and slopes Total Trails (km) Total Slopes (km) Total distance (km) 73.89 91.66 165.550
Team 7416 HiMCM 2017 Page 22 of 24 Table 8. This table shows given data
Team 7416 HiMCM 2017 Page 23 of 24 Table 9. This table shows the ranking system
Team 7416 HiMCM 2017 Page 24 of 24 References Breckenridge Reviews & Ratings. (n.d.). Retrieved November 10, 2017, from https://www.onthesnow.com/colorado/breckenridge/reviews.html Skiresort.info The largest ski resort test portal in the world. (n.d.). Retrieved November 10, 2017, from http://www.skiresort.info/ Wasatch Peaks Ranch Mirr Ranch Group. (n.d.). Retrieved November 09, 2017, from https://www.mirrranchgroup.com/ranches/wasatch-peaks-ranch/ Worldwide. (n.d.). Retrieved November 10, 2017, from http://www.skiresort.info/ski-resort