Team 7414 HiMCM 2017 1/12 Ski Slope Problem Wasatch Peaks Ranch in Peterson, Utah is for sale, and a group of wealthy winter sport fans and their agent, Ms. Mogul, are looking for a new mountain to develop into a ski resort that could possibly host the Winter Olympics in the future. The ranch, of nearly 13 000 acres, has around 5500 potential acres of skiable area. We were tasked with developing the ranch area into one of the top ski resorts in North America and a potential future Winter Olympics location. Summary The Winter Olympics is a time for athletes to showcase their skills and expertise. Years of hard work are displayed on a screen, or even better, in person. There is no better time for a city or country to prosper from these international games. What makes Wasatch Mountain any different? As the athletes come with their friends, family, and fans, a place such as Salt Lake City is elevated by the influx of people and money. That being said, the host city must be prepared for the rapid increase of people and create an arena that will be suitable for the size of the crowd, events, and other commodities. We created our model to meet two standards: the best resort in North America and the ideal location of the next Winter Olympics. In order to evaluate our model, we established criteria to compare a sample of North American ski resorts to Wasatch Mountain. These criteria include, but are not limited to, peak height, area for spectators, annual snowfall, trail variations, and number of chairlifts. Our model takes these variables into account and utilizes topographi! cal maps, tables, and ski resort statistics. By analyzing the terrain of Wasatch Mountain, we made a model to position trails, which vary in length and difficulty, and created a system of calculating the optimal number of chairlifts. For part I, our initial strategy was to use the topographical and satellite maps to determine the percent slopes of specific areas on the map. To more efficiently find the percent slopes, we created a Java program using values of the elevation and horizontal distance from the lodge at a certain point. On the map, we noticed that as the elevation decreased, the slope of the mountain decreased, as the levels on the topographical map got further apart. We then found the total number of trails. We used the average width of a trail, calculated using the data of an accurate sample of resorts located in North America, and then divided the skiable area by this designated width. This length was used to determine the total number of trails. In part II, we evaluated our model using set criteria and those criteria in comparison to other resorts in North America. We also looked at ratings of the North American resorts provided to justify elements of our model that are superior to surrounding resorts. Customer ratings and reviews were used to evaluate which of the resorts' factors were important to the customers and which aspects of the resort were unsatisfactory. Analyzing ratings allowed us to take the input of customers at other resorts and integrate it into our own resort. One weakness of our model is that many assumptions and estimates had to be made, some without much information. In addition, we only have example runs, not an entire map. The topographic map did not allow for specified heights, therefore our calculations may be unreliable. Also, there was a low sample size for the ratings of the resorts, which may slightly affect the data. A strength of our model is that we took into account many, varying factors that would significantly affect the outcome of the resort. Some of the main factors include peak height of the mountain, annual snowfall, and number of chairlifts.
Team 7414 HiMCM 2017 2/12 MEMO To: Ms. Mogul From: 7414 Date:11/10/17 Re: Ski Slope Proposal Just thirty-five minutes from Salt Lake City, Utah and its international airport, Wasatch Mountain offers a beautiful resort with numerous commodities and practical logistics. Wasatch Mountain is about 13,000 acres used for skiing and other winter activities. The mountain has twenty-four peaks and fifteen bowls, with a 4,750-foot drop between the average peak and bowl. The mountain also has an eleven-mile ridgeline. Before evaluating the region, criteria regarding elevation, skiable area, number of trails, and number of chairlifts, were established. These criteria help to form more reliable comparisons of the Wasatch Mountain to the other North American ski resorts. The elevation of the highest peak of Wasatch Mountain surpasses all of the highest peaks of all the other North American resorts. Realizing that the difference between a mountain s peak elevation and its base elevation is more important than just the peak elevation, the elevation of the skiable area was determined (column labeled Elevations ). The elevation of Wasatch Mountain s skiable area remains the maximum elevation after these calculations are done. Table 1: Peak Elevation. Red represents the maximum values(s), while dark green represents the minimum value(s). The area for spectators is 2,000 acres more than the skiable area. The 7,500 acres of unoccupied land will provide space for sectors. The Olympic stadium can hold roughly 75,000 people, therefore, for theoretical analysis, we can pretend that 75,000 spectators will watch the Olympics in person, as this will most likely be the maximum amount of people in attendance. If this is the case, in theory, each person will have 0.1 km to themselves. It can be assumed that this would not happen in reality, however, the numbers convey the abundant land for the spectators and athletes. The annual snowfall for each of the specified locations in North America was found and compared to the annual snowfall for Salt Lake City, Utah since the mountain is very close to this location. Although the annual snowfall for Wasatch Mountain was not the highest compared to the other locations, when the average of all the snowfall is found, the annual snowfall for Wasatch Mountain is 77.94 in above average. This means that the snow is fresher than the average North American resort. For this, less snow will have to be made and the quality of the trails will be superior than the average North American trail.
Team 7414 HiMCM 2017 3/12 Table 2: Annual Snowfall. The final number [322.06] is an average of the annual snowfall for all of the North American locations. Wasatch Mountain has a variety of trails for all levels. Although this does not pertain to the Olympics, the reputation of satisfying all skill levels will create a great relationship with the customers. More people will be inclined to attend the Olympics at a location resort they support. The mountain offers thirty-seven green circle trails, seventy-three blue square, and seventy-three black diamond trails. The total length of all the trails combined, 236.784 km, offers customers with the option to attempt many different trails. the total length can be used to find the amount of the total length of each skill level. For instance, 20% of the trails are green squares, 47.3568 km, and the blue square trails and black diamond trails each make up 40% of the total amount of trails. The 183 trails give people a combined 236.784 km of varying difficulty. The number of chair lifts needed at Wasatch Mountain is based on the need to decrease crowding and decreasing the cost. As the number of lifts increase, the cost increases, while the crowding decreases. Wasatch Mountain has 24 lifts. Decreasing the crowds is more important than decreasing the cost. For this reason, by averaging the data of the lowest amounts of trails per lift, the average number of lifts needed to have a minimal amount of crowding is determined. This method is flawed, as it is not possible to calculate an exact number of people who will come to Wasatch Mountain; therefore, the number of chairlifts must be determined based on the ratio of trails to lifts at the other resorts in North American. By referring to these averages, the model still represents a theoretical number of chairlifts that would result in the least amount of crowding. Table 3: Trails per Lift.
Team 7414 HiMCM 2017 4/12 Initial Data Name State Country Elevation Peak (m) Base Elevation (m) Acres Skiable Green km Blue km Black km Beaver Creek Colorado USA 3488 2255 1832 28.5 64.5 57 16 Big Resort Sky Montana USA 3398 2072 5800 55 69 126 28 Breckenridge Colorado USA 3914 2926 2908 28 60 65 23 Breckenridge British Columbia Canada 2134 1052 2500 42 58 42 9 Hole Jackson Wyoming USA 3185 1924 2500 16 50 50 12 Killington Vermont USA 1285 355 1509 37.4 43 46 20 Lake Louis Alberta Canada 2637 1646 4200 35 62 42 7 MountainPark City Utah USA 3029 2080 7300 27 152 71 38 Silver Star British Columbia Canada 1915 1155 3269 20 50 45 12 Squaw Valley California USA 2760 1890 3600 25 45 30 24 Steamboat Springs Colorado USA 3221 2103 2956 25 95 45 17 Sugarloaf Mountain Maine USA 1286 426 1153 28 40 51 13 Sun Peaks British Columbia Canada 2082 1198 4270 13.5 78 43.5 9 Vail Colorado USA 3433 2457 5289 57 84 93 25 Whistler Blackcomb British Columbia Canada 2284 653 8171 40 110 50 26 Winter Park Resort Colorado USA 3676 2743 3000 11 53 79 22 Initial Data with Calculated Data Lifts Name Acres Skiable Skiable Areas in km^2 Slopes Total km Green (%) Blue (%) Black (%) Lifts Total Number of Trails Average Width of Trail Beaver Creek 1832 7.4138 150 19.00 43.00 38.00 16 150 0.0494 Big Resort Sky 5800 23.4718 250 22.00 27.60 50.40 28 306 0.0939 Breckenridge 2908 11.7683 153 18.30 39.22 42.48 23 187 0.0769 Breckenridge 2500 10.1172 142 29.58 40.85 29.58 9 133 0.0712 Hole Jackson 2500 10.1172 116 13.79 43.10 43.10 12 155 0.0872 Killington 1509 6.1067 126.4 29.59 34.02 36.39 20 145 0.0483 Lake Louis 4200 16.9968 139 25.18 44.60 30.22 7 337 0.1223 Mountain Park City 7300 29.5421 250 10.80 60.80 28.40 38 132 0.1182 Silver Star 3269 13.2292 115 17.39 43.48 39.13 12 170 0.1150 Squaw Valley 3600 14.5687 100 25.00 45.00 30.00 24 165 0.1457 Steamboat Springs 2956 11.9625 165 15.15 57.58 27.27 17 163 0.0725 Sugarloaf Mountain 1153 4.6660 119 23.53 33.61 42.86 13 135 0.0392 Sun Peaks 4270 17.2801 135 10.00 57.78 32.22 9 195 0.1280 Vail 5289 21.4038 234 24.36 35.90 39.74 25 200 0.0915 Whistler Blackcomb 8171 33.0669 200 20.00 55.00 25.00 26 166 0.1653 Winter Park Resort 3000 12.1406 143 7.69 37.06 55.24 22 183 0.0849
Team 7414 HiMCM 2017 5/12 Givens - 13,000 acre ranch - Estimated 5,500 acres of potential ski slopes - An 11 mile ridgeline - A 4750 foot drop among its 24 peaks - 15 bowls Criteria - Plenty of runs of varying lengths - A total of at least 160 km of slopes - 20% beginner slopes (green circle), 40% intermediate (blue square), and 40% difficult (black diamond) Assumptions - If the mountain is chosen for the Olympics, the trails can be modified to suit each of the events. - Skiable area takes into account the roads that cannot be skied upon. - The mountain is steeper and narrower as it approaches the peak, increasing difficulty. - The skiable area is used entirely, and takes into account the elimination of dangerous hazards such as rivers and large cliffs. - The sample of data represents a variety well-reviewed North American resorts. - The fresher the snow, the better the condition of the trail. Strategy Our initial strategy was to use the topographical and satellite maps to determine the percent slopes of specific areas on the map. To more efficiently find the percent slopes, we created a program that would use values of the elevation at a certain point and the horizontal distance to calculate percent slope. On the map, we noticed that as the elevation decreased the slope of the mountain decreased, because the rings on the topographical map got further apart. However, we came to the conclusion that the maps were unreliable because the elevation for a specific point could not be found, meaning that there was a lot of room for error. We modified our strategy to calculating the number of runs we needed, the total lengths of these runs, and the difficulty ratings of these runs to get a baseline for the design of our resort. The initial data and the research we conducted was used to make the most accurate estimates and assumptions, which were then compared to each other. Customer ratings and reviews were used to evaluate which of the resorts factors were important to the customers, and which aspects of the resort were unsatisfactory. This allowed us to take the input of customers at other resorts and apply it in making our own resort. We also calculated the number of lifts we would need at our resort, as having too few lifts causes overcrowding and impacts the resort s ability to have a top ranking in North America.
Team 7414 HiMCM 2017 6/12 Solution Initially, the topographic and satellite maps were used in an attempt to determine the percent slopes of specific areas on the map. The map was simple to use, but did not have features that allowed exact measurements, therefore making slope calculations complex and creating a potential for a large percent error. It was observed that as the elevation decreased on the map, the rings on the topographical map got further apart, so the slope of the mountain decreased. This is the basis for our assumption that the mountain is steeper and narrower as it approaches the peak. Also, it was not specified which units the numbers on the map were in. In order to more efficiently find the percent slopes, we created a simple Java program (see Appendix) that would take two inputs: the values of the elevation at a certain point and the horizontal distance, to calculate percent slope. This would have made calculations simpler, but due to the complexity of the topographic map, we did not end up using the program or performing these calculations until our strategy was complete. To convert the skiable area to kilometers squared, we multiplied the total skiable area in acres, 5500, by the conversion factor of 0.00404686. The new number, 22.2577 is the skiable area in units of kilometers squared. For each location, the skiable area is divided by the total length of all the location s runs added together. For example, In Beaver Creek (see Table 1), the skiable area is 7.4138 kilometers squared, and the sum of the lengths of all the runs is 150 kilometers. 7.4138 / 150 = 0.0494, so the average width of a run in Beaver Creek is 0.0494 kilometers. The quotient provided us with the average width of the runs for the said location. After all of these widths were calculated, we took the average to find the average width of a run. There was a large range of different widths, so without confirmed data, our best estimate was the average. We found that the average width is 0.0943, and used this average width as the average width for the runs at Wasatch Mountain. To find the total length of the runs at Wasatch Mountain combined, we divided the skiable area of 22.2577 kilometers squared by the average width 0.0943 km and got a total length of 235.907 km. The sum of the lengths of all the runs in our ski resort is 235.907 km. Another way to phrase this is that with the skiable area, we used the total number of runs in the ski resort to find the average area of a run. The skiable area divided by the sum of lengths of the runs will also give us the average width of a run. This calculation evenly distributes the area over the total length. Therefore, the average width of each run, TW, is the skiable area or AS, 22.26 km squared, divided by the sum of lengths of the runs, LS, 236.78 km which results in a value of 0.0943 km for TW. Given this, we moved on to calculate the average length of a runs. First, we found the average area of a run by dividing the skiable area by the total number of runs. The resulting area of each run, AT, is 0.1216 km squared. With the average width of each trial, TW, and the average area of each run, AT, we calculated the average length of each run, AL; AT divide by TW is 1.2939 km. To verify the sum of the lengths of the runs, LS, is roughly 236.7 km.
Team 7414 HiMCM 2017 7/12 Table 1 We researched the total number of runs or runs that a ski resort had for the given list of resorts (see Table 2). Then, we averaged these numbers to determine an acceptable baseline to calculate the average area of the run. Table 2 Here, we show that the irregular shape of the skiable area can be translated into a rectangular area with hypothesized dimensions for width and lengths. For example, let us establish the total area of some arbitrary shape (see Figure 1) to be 100 square units. With this area, we can change create an average shape that is defined by a specific length to width ratio, that has a length of 20 and a width of 5. Similarly, we can take the abstract shape of given skiable area and show that it can be translated into a rectangular region.
Team 7414 HiMCM 2017 8/12 Figure 1: Abstract Representation of Wasatch Mountain Area Based on our assumption that the terrain becomes more difficult with elevation, we decided that the starting height of a run is proportionate to the difficulty of the run. For example, a black diamond would most likely begin much higher on the mountain that a green square, because the percent slope is higher there. In addition, some runs may lead down into another run of equal or lesser rank, but the next run is not added onto the length of the run above it. For example, the length of a black diamond run that leads into a blue square is not the length of the two combined, it is just the length of the black diamond run itself. Based on that information, we decided to split the total length of all the runs combined between the difficulties 20% to 40% to 40% ratios. The total length of the green circles in Wasatch Mountain run is 47.36 km, the total length of the blue squares is 94.71 km, and the average length of the black diamonds is 94.71 km. Table 3 In order to evaluate customer satisfaction, we researched the ratings (out of five stars) of the provided resorts, to determine which of the resorts were determined to be high quality. The average of the ratings was around 3.9 stars, but ranged from 3.5 to 4.5 stars. We also found ratings about specific aspects of the resorts, such as how uncrowded and child friendly they were, along with ratings of run and lift quality. We used the same website for as many of the resorts as we could. While the sample size was small, the opinions of people who have visited these resorts were important in making our resort one of the top in North America. We inserted a column with the ratios of trails to the number of lifts by dividing by the number trails by the number of lifts in each respective location (see Table 4). The average established a base value of ratio of runs to lifts, which corresponds to the amount of crowding. However, we wanted to
Team 7414 HiMCM 2017 9/12 value of ratio of runs to lifts, which corresponds to the amount of crowding. However, we wanted to create a resort with the least amount of crowding, so we decided to divide the trails to lifts ratios in levels of three to establish a high crowding environment, a medium crowding environment, and a low crowding environment. We decided to find the optimum trails to lifts ratio, so we averaged the values from the lowest crowding level. Our trails to lifts ratio was found to be 7.65. To determine the total number of lifts needed in the park, we divided the ratio by the total number of trails and got a rounded value of 24 chair lifts total. Table 4 Table 5 Rise (ft) Run (ft) Percent Slope Beginner Slope 4809.94 19891.57 24.18 % Intermediate Slope 5319.26 15533.76 34.24 % Advanced 8094.94 8155.42 99.24 % Above (see Table 5) are some late calculations we did to find three runs, one for each difficulty level. The percent slopes are what define the slopes as beginner, intermediate, and advanced. We used the rises and runs to calculate the percent slope using our Java program (see Appendix). Weaknesses - Many assumptions and estimates had to be made, some without much information. - We only have example runs, not an entire map. - The topographic map did not allow for specified heights. - There was a low sample size for the ratings of the resorts.
Team 7414 HiMCM 2017 10/12 Conclusions Our total number of runs was 183, which is equal to the average number of trails for the resorts we were given. More runs does not always equal better runs. There are 37 green circle trails, and both blue square and black diamond have 73 trails each, for a total of 183. The lengths and widths of these trails can vary greatly, but they will always average the same. Many of our trails run parallel to each other, to provide more trail options from certain areas. There are approximately 7 or 8 runs per lift, and we have 24 lifts, providing access to all parts of the mountain. This allows limited crowding, which is good for the Olympics because it is a highly viewed event. Our resort is close to Salt Lake City, a major city, which allows easy access to other parts of the country, and the world, countries with its International Airport. In conclusion, our resort can be considered one of the top in North America, and is a valid contender for the Olympics. Extensions Cost Roads In our solution, we did not consider money and its implications in the problem, as the problem description mentioned that wealthy fans of winter sports would be building the resort. With additional time to extend our solution, we would have taken into consideration the cost to build all the runs and all the chairlifts. These values could have been used to figure out a rough estimate for ticket price at our resort. While modeling the scenario, we assumed that roads passing through the ski trails were not included in the skiable acres data, prohibiting us from finding the exact locations of each run. To provide a more complete solution in the future, we would plot each run. By using the locations of the runs, we would have figured out which roads intersect with which runs. Determining where these intersections would have allowed us to specify which traves would have shortened lengths due to the road. Evaluating Topographical and Satellite Maps With extra time, we would have created a strategy that included elevation at specific locations on the mountain. This strategy would have allowed us to look at elevation around the skiable area and use our topographical map program to sample elevations, and determine steepness using our Java program. Then, we would draw theoretical slopes of varying difficulty (green, blue, and black) on the map.
Team 7414 HiMCM 2017 11/12 Bibliography Blackcomb, W. (2014). The Mountains - Skiing & Snowboarding. Retrieved November 10, 2017, from http://ww1.whistlerblackcomb.com/media/stats/mountains/ Consult, M. (n.d.). Size of the skiing area. Retrieved November 10, 2017, from http://www.pistenlaengen.- co3m/en/ski-resort-size.html Killington Resort Reviews & Ratings. (1995-2017 ). Retrieved November 10, 2017, from https://www.onthesnow.com/vermont/killington-resort/reviews.html Killington Resort Reviews & Ratings OnTheSnow www.onthesnow.com Killington Ski Resort. (2017, October 30). Retrieved November 10, 2017, from https://en.wikipedia.org/wiki/killington_ski_resort Killington Ski Resort - Wikipedia en.wikipedia.org... Mirr, K. (n.d.). Wasatch Peaks Ranch. Retrieved November 10, 2017, from https://mapright.com/ranching/maps/a9d794878e9aad24786a35fbdf4057af/share Mountain Info. (2017). Retrieved November 10, 2017, from https://www.beavercreek.com/the-mountain/about-the-mountain/mountain-info.aspx Mountain Stats. (2017). Retrieved November 10, 2017, from https://www.jacksonhole.com/mountainstats.html Mountain Stats. (2017). Retrieved November 10, 2017, from https://www.steamboat.com/the-mountain/- mountain-stats Mountain Statistics. (2009-2017). Retrieved November 10, 2017, from https://www.skilouise.com/themountain/mountain-stats.ph Sports America Inc. (2014). Retrieved November 10, 2017, from http://www.sportsamerica.com/skiing/squawvalley_stats.php Stats and Facts. (2017). Retrieved November 10, 2017, from http://sugarloaf.com/media-room/resortstats Terrain and Lift Status. (2017). Retrieved November 10, 2017, from https://www.vail.com/the-mountain/- mountain-conditions/terrain-and-lift-status.aspx Terrain & Lift Status. (n.d.). Retrieved November 10, 2017, from https://www.parkcitymountain.com/themountain/mountain-conditions/terrain-and-lift-status.aspx
Team 7414 HiMCM 2017 12/12 mountain/mountain-conditions/terrain-and-lift-status.aspx Trail Maps & Stats. (n.d.). Retrieved November 10, 2017, from https://www.sunpeaksresort.com/skiride/the-mountain/trail-maps-stats Top Rated in USA. (n.d.). Retrieved November 10, 2017, from http://www.powderhounds.com/usa Wasatch Peaks Ranch Mirr Ranch Group. (n.d.). Retrieved November 10, 2017, from https://www.mirrranchgroup.com/ranches/wasatch-peaks-ranch/#prop-maps Winter Park Resort. (2017, August 18). Retrieved November 10, 2017, from https://en.wikipedia.org/wiki/winter_park_resort Z. (2017). Beaver Creek Resort Luxurious Ski Trips, Famous Chocolate Chip Cookies. Retrieved November 10, 2017, from https://www.zrankings.com/ski-resorts/17-beaver-creek-resort Appendix Java Code import java.util.scanner; public class HIMCM { public static void main(string[] args) { Scanner sc = new Scanner(System.in); System.out.println( Type in rise and run values. ); double rise = sc.nextdouble(); double run = sc.nextdouble(); String leveldifficulty = ; double percentslope = (rise/run)*100; if(percentslope < 6) { leveldifficulty = not a valid level ; } else if (percentslope >= 6 && percentslope < 25) { leveldifficulty = beginner slope ; } else if (percentslope >= 25 && percentslope < 50) { leveldifficulty = intermediate slope ; } else { leveldifficulty = advanced slope ; } System.out.println( The percent slope is + percentslope +. The level of difficulty is + leveldifficulty+. ); } }