The Fastest Cyclist

8078 /www/nrich/html/content/id/8078/ 1 0000-00-00T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><mdo:image src="cyclists.png" style="float: right;"></mdo:image> In <a href="/7571">Nutrition and Cycling</a>, you were invited to work out Andy's calorie needs on a ride from Land's End to John o'Groats. If you haven't already done so, print out a copy of <a class="pdflink" href="/content/id/8078/Nutrition%20and%20Cycling%20cards.pdf">these cards</a>. Andy is competing in a Land's End to John o'Groats race, and YOU are his coach! You are determined that Andy will win the race, as long as he is willing to cycle for more than 7 hours a day. What is the shortest time in which Andy could complete the ride, if we ignore his 7 hour daily cycling limit, while making sure his energy needs are met? You may wish to consider minimising the cost, and/or the weight that Andy needs to carry, together with any other appropriate constraints that occur to you. <div class="framework">NOTES AND BACKGROUND: Sport nutrition is a specialist field, and there are hundreds of products such as drinks, gels and bars for an endurance athlete to choose from. In reality, there are more factors to consider than calorific intake, and a cyclist cneeds to balance carbohydrate intake for basic energy, protein intake for muscle building, muscle glycogen reserves, maximising recovery speed and keeping hydrated: all important factors to consider. You might find <a href="http://www.lifeslittlemysteries.com/what-do-tour-de-france-riders-eat-0912/">this article</a>, describing the diet of Tour de France cyclists, of interest. Cycling image from <a href="http://commons.wikimedia.org/wiki/File:EVD-ciclismoruta-768.jpg">Wikimedia Commons</a>, uploaded by <a href="http://commons.wikimedia.org/wiki/User:Evdcoldeportes">Evdcoldeportes</a>, released under the Creative Commons <a href="http://creativecommons.org/licenses/by-sa/2.5/co/deed.en">Attribution-Share Alike 2.5 Colombia</a> license.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> This problem offers a challenging extension to <a href="/7571">Nutrition and Cycling</a>. Whereas the former problem requires students to make sense of information and engage in proportional reasoning, this is an optimisation task which students could tackle using spreadsheets. <h3> Possible approach</h3> Students will need to have worked on <a href="/7571">Nutrition and Cycling</a> first, and will need the same <a class="pdflink" href="/content/id/8078/Nutrition%20and%20Cycling%20cards.pdf">set of cards</a>. "What if Andy could cycle for longer than seven hours a day? Imagine you are his coach, and you need to come up with a plan that makes sure his energy needs are met, but helps him to complete the race in the fastest possible time. At the end of the task, you will each get a chance to present your race plan to the rest of the group. Who can come up with the winning plan?" Arrange the class in small groups. Students will need access to calculators or spreadsheets. While you circulate, ensure that students are recording their calculations clearly so that they will be ready to present them. There is no 'correct' answer to this task - students will have to make decisions about whether a calorie deficit can be permitted, and whether additional off-road snacks could be allowed. Plenty of time should be allowed at the end of the task for groups to present their plans, the decisions they made, and their reasons, with clear justifications for their results. <h3>Key questions</h3> What are Andy's calorie needs? What are Andy's calorie allowances? How do these change as he cycles for a longer time? <h3>Possible extension</h3> Students may wish to represent the situation algebraically, and apply different constraints before solving the resulting equations. <h3>Possible support</h3> Encourage students to use trial and improvement - choose a time greater than 7 hours for Andy to cycle, calculate his calorie intake and calorie needs, and then adjust up or down as necessary.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">What is the maximum calorie intake available to Andy, based on the snacks and energy drinks he can carry? If he cycled for $7\frac{1}{2}$ hours, how many calories would he need? How many calories could he take in? What if he cycled for $8$ hours?</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">This spreadsheet shows the limitations assuming all other information stays the same when the 7 hour per day limit is relaxed: <a href="/content/id/8078/Fastest%20Cyclist.xls">Fastest Cyclist.xls</a> Leads to the conclusion that 8 hours per day cycling would give him a very small calorie deficit. Algebraic solution (where h = hours/day cycling): Calories available to consume: 4000 + 350h Calories used: 2500 + (14 x 1.61 x 100h)/4.19 4000 + 350h = 2500 + (14 x 1.61 x 100h)/4.19 h = 7.98 hours per day cycling If Andy cycles 8 hours per day, total race time: 1407 / (1.61 x 14 x 8) = 7.8 days = 7 days + 6.422 hours cycling on Day 8 = 7 days + 6 hours 25.34 minutes cycling on Day 8 Calculations below relate to Yanqing's original version of the problem. Cycling calories = 3750 a day, so total 6250 calories a day. (rounded) Meals 3000 3250 remaining to be made up from: Off-road snacks up to 1000 On-road snacks up to 2450. Gives an extra 200. Bananas 120, 12kcal for 1p, 1kcal for 1g, holds 8 Cereal bar 100, 6.25kcal for 1p, 4kcal for 1g, holds 16 Gel 110, 1.1 kcal for 1p, 2.6kcal for 1g, holds 24 energy bar 220, 2.2kcal for 1p, 3.4kcal for 1g, holds 8 drink 190, 1.3 kcal for 1p, holds 6. Possible plans: Meals, full off-road snacks, 8 energy bars, 2 (and a bit) servings of drink Meals, 24 packets of gel, 850 kcal of snacks Meals, 16 cereal bars, 3 (and a bit) servings of drink, full snacks For cheapest and lightest, sensible to maximise off-road snacks. Cheap: 16 cereal bars, 3 and a bit servings of drink costs about £7.30 a day. Light: topping up water the full 4 times, start with 1 bottle (2 servings) of energy drink, then 8 energy bars. --- Going for 7.85 hours for 8 days requires 2700kcal on the road per day, doable but must use lots of gel. Now going for 9 hours for 7 days: 7300 kcal a day, so 3300 for on the road! That's over the limit for energy absorption, so he will need to eat more off the road. </mdoxml> 2 4 0 0 0 1 0 The Fastest Cyclist Andy is desperate to reach John o'Groats first. Can you devise the winning cycling and nutrition plan? Decision Mathematics and Combinatorics Optimisation