problem a paper: swim, bike, and run - wpiusers.wpi.edu/~ncng/images/himcm.pdf · team 6285 himcm...

Team 6285 HiMCM 2016 of 19

Problem A Paper: Swim, Bike, and Run

The mayor of a town wants to support a local youth group by hosting a triathlon.

Participants must swim 1500 m, bike 40 km, and run 10 km. Racers must transition between

each stage. There will be a wide variety of skill levels, ranging from professional to amateur.

About 2000 contestants are expected to attend, but a small buffer of five percent was added to

the expected attendance for safe measure. Several triathlons have divisions for the athletes based

on gender, age, and skill level. People will compete against other runners in their division but

will not always run with them. These divisions will yield multiple winners from the 2000

expected participants. Additionally, the mayor wants to minimize the course congestion and the

time spent on the roads because the roads can only be closed for five and a half hours. The team

was tasked with making a triathlon by examining collections of data from previous triathlons.

The team found that the participant count per wave is 50 people. The average time interval in

between waves of contestants will be three minutes. Using this information, the team constructed

43 waves consisting of mixed genders, ages, and experience, but common previous final times.

The wave of the slowest contestants would begin and after a three minute intervals the next wave

of people would begin. Both a scatter plot and a box and whisker plot were constructed which

suggest that there will be no major clogging of the streets. The team was also told to find

advantages in changing the distances of the triathlon. An equation that defined velocity as a

function as time was used to account for decreases in stamina. The results indicate that the best

possible solution for minimizing the road closure time and course congestion is to shorten the

distances of cycling and running and leave the swimming section constant.


Dear Mayor Lawrence,

The triathlon that you requested will be a great success! As of now, we are expecting

approximately 2000 participants, but there is a possibility that more people may attend than

expected. Therefore, the triathlon has been designed to accommodate a five percent increase in

the number of expected athletes, allowing for an additional 100 competitors. To establish a

competition between the participants, divisions of contestants have been devised based on

gender, age, and skill level. Participants have also been divided into separate waves indicating

starting times. Participants will compete with people in only the same division, not necessarily

the same wave. In order to minimize course congestion, the participants were divided into 43

separate waves based on their final time results from a previous triathlon. The first wave will

start the course at 12:00 p.m. and will be followed by a new wave every three minutes.

Moreover, the time delay will lower the amount of course congestion by allowing leading

athletes to distance themselves from succeeding groups. Creating distance between participants

will effectively make the course safer by preventing collisions. If the roads are closed to the

public by 12:28 p.m., then the town can reopen local roads exactly five hours and thirty minutes

after the roads were blocked off. Before 12:28 p.m., the expected participants will not have

reached the cycling stage yet; the participants will not have completed transition one and

therefore, the roads may remain open to the public. At 5:58 p.m., the roads should be once again

opened to the public. Ultimately, the decreasing of congestion and the ability to reopen roads in

minimal time will lead to a successful event. The corresponding schedule is attached.

Best Regards

(The Schedule is attached at the end of the appendix)

Team 6285 HiMCM 2016 of 19 Triathlon Day Schedule

Sponsored by Super Tread Race Company ©

Roads are Closed to Public by: 12:28:00 PM

Roads are Opened by: 5:58:00 PM

Time Previous Time Time Start Time

Time of Final ≥ 4:13:37 12:00:00 PM

4:13:37 > Time of Final ≥ 4:00:11 12:03:00 PM

4:00:11 > Time of Final ≥ 3:50:47 12:06:00 PM

3:50:47 > Time of Final ≥ 3:44:25 12:09:00 PM

3:44:25 > Time of Final ≥ 3:39:05 12:12:00 PM

3:39:05 > Time of Final ≥ 3:33:59 12:15:00 PM

3:33:59 > Time of Final ≥ 3:29:58 12:18:00 PM

3:29:58 > Time of Final ≥ 3:27:08 12:21:00 PM

3:27:08 > Time of Final ≥ 3:23:44 12:24:00 PM

3:23:44 > Time of Final ≥ 3:21:17 12:27:00 PM

3:21:17 > Time of Final ≥ 3:18:15 12:30:00 PM

3:18:15 > Time of Final ≥ 3:16:02 12:33:00 PM

3:16:02 > Time of Final ≥ 3:13:56 12:36:00 PM

3:13:56 > Time of Final ≥ 3:11:27 12:39:00 PM

3:11:27 > Time of Final ≥ 3:09:31 12:42:00 PM

3:09:31 > Time of Final ≥ 3:07:25 12:45:00 PM

3:07:25 > Time of Final ≥ 3:05:22 12:48:00 PM

3:05:22 > Time of Final ≥ 3:03:44 12:51:00 PM

3:03:44 > Time of Final ≥ 3:01:52 12:54:00 PM

3:01:52 > Time of Final ≥ 3:00:13 12:57:00 PM

3:00:13 > Time of Final ≥ 2:58:34 1:00:00 PM

2:58:34 > Time of Final ≥ 2:56:55 1:03:00 PM

2:56:55 > Time of Final ≥ 2:55:11 1:06:00 PM

2:55:11 > Time of Final ≥ 2:53:28 1:09:00 PM

2:53:28 > Time of Final ≥ 2:52:05 1:12:00 PM

2:52:05 > Time of Final ≥ 2:50:26 1:15:00 PM

2:50:26 > Time of Final ≥ 2:48:35 1:18:00 PM

2:48:35 > Time of Final ≥ 2:47:06 1:21:00 PM

2:47:06 > Time of Final ≥ 2:45:41 1:24:00 PM

2:45:41 > Time of Final ≥ 2:44:20 1:27:00 PM

2:44:20 > Time of Final ≥ 2:42:49 1:30:00 PM

2:42:49 > Time of Final ≥ 2:41:21 1:33:00 PM

2:41:21 > Time of Final ≥ 2:39:46 1:36:00 PM

2:39:46 > Time of Final ≥ 2:37:33 1:39:00 PM

2:37:33 > Time of Final ≥ 2:35:19 1:42:00 PM


Given Parameters

● About 2000 people are expected to attend the event

● The triathlon consists of 1,500 m of swimming, 40,000 m or cycling, and 10,000 m of

running.

● Local roads may not be closed for more than five hours and thirty minutes.

Additional Assumptions:

● All professional and premier contestants will attend the event and will arrive on time.

● There are approximately 50 participants per wave.

● Equation A (See appendix) correctly represents both biking and running situations.

● The length of the swimming portion remains constant, as one cannot change the length of

a lake (part 2).

● The times given in the results of the recent triathlon will be approximately equal to the

times in this triathlon.

● The highest percent increase in the number of contestants is 5%.

● The roads do not have to be closed for the swimming or the transition 1 stage.

METHODOLOGY

2:35:19 > Time of Final ≥ 2:33:09 1:45:00 PM

2:33:09 > Time of Final ≥ 2:30:36 1:48:00 PM

2:30:36 > Time of Final ≥ 2:28:11 1:51:00 PM

2:28:11 > Time of Final ≥ 2:24:21 1:54:00 PM

2:24:21 > Time of Final ≥ 2:20:05 1:57:00 PM

2:20:05 > Time of Final ≥ 2:11:09 2:00:00 PM

2:11:09 > Time of Final ≥ 1:48:32 2:03:00 PM

1:48:32 > Time of Final 2:06:00 PM


Data Analysis:

Determining the Order of the Contestants:

Contestants were divided into waves based on the contestant’s previous triathlon final

time such that contestants with similar times were grouped together into waves of

approximately 50 participants. A few exceptions were made to the fifty participants in a

wave. For example, there were not enough remaining contestants after creating the

groups to form a full fifty person wave.

Determination of Divisions:

The divisions were determined by the category each racer was placed in, their gender,

and their age. First, the the racers were divided by gender. Each was then split into three

groups: the heavyweight category (Clydesdales or Athenes), the open category, and the

competitive category (Premiers and Pros). The open category was also subdivided based

on age groups with the following age ranges making seven divisions: 19-28, 29-32, 33-

37, 38-41, 42-46, 47-50, and 51 and above. The average number of contestants in an open

division was 307 for males and 128 for women. In total there are 18 divisions, 9 for each

gender.

Method for Determining Open Divisions:

The team decided that in order to keep open divisions at a competitive size, there

should be seven divisions. First, a count of the number of participants from each

age was recorded and a graph was made. Then, the genders were split up and a


graph of the number of participants for each age was made for each gender as

shown below.


y = -3E-06x5 + 0.0006x4 - 0.0478x3 + 1.1608x2 + 7.9715x - 337.37R² = 0.9524

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Age, years

Participants v. Age (Total)

y = -3E-06x5 + 0.0006x4 - 0.0545x3 + 1.8861x2 - 19.871x - 16.968R² = 0.957

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Age, years

Participant v. Age (Male)


Six divisions were made out of the youngest nine age groups that were not in a

division already and the last one was made out of the remaining seven age groups.

For example, the first division was consisted age groups 19 - 27, the second

division consisted of age groups 28-36, etc. The age groups were rearranged so

that the standard deviation for each model was minimized. This method was done

for the male group first and then the same was done for the female group. The

arrangement of the age groups in the divisions based on minimizing the standard

deviation for the male divisions will be called Method 1. The arrangement based

on the female divisions will be called Method 2. In order to keep the age ranges

the same for both the males and the females the team used Method 1 to group the

females and Method 2 to group the males. This data was organized into the tables

y = -2E-07x5 - 4E-06x4 + 0.0067x3 - 0.7253x2 + 27.842x - 320.4R² = 0.871

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Age, years

Participants v. Age (Female)


and graphs below. In the graphs, the blue dotted line is the line of best fit for the

data and the orange dotted line is the average for the set of data.

Method 1

Age

Ranges F M

Number of Participants

(P) P/2147

Number of Paticipants

(P) P/2147

19-29 157 0.174833 282 0.131346

30-34 209 0.232739 362 0.168607

35-39 145 0.16147 385 0.17932

40-44 132 0.146993 356 0.165813

45-49 118 0.131403 283 0.131812

50-54 76 0.084633 220 0.102469

55+ 61 0.067929 259 0.120633

Average 128.29 306.71

STDEV 46.29 56.80


Method 2

Age

Ranges F M

Number of Participants

(P) P/2147

Number of Paticipants

(P) P/2147

19-29 157 0.174833 282 0.131346

30-34 209 0.232739 362 0.168607

35-39 145 0.16147 385 0.17932

40-44 132 0.146993 356 0.165813

45-49 118 0.131403 283 0.131812

50-54 76 0.084633 220 0.102469

55+ 61 0.067929 259 0.120633

Average 128.29 306.71

STDEV 28.79 57.90


R² = 0.9634

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Open Division Number

Participants v. Group Number (Male - Method 1)

R² = 0.8902

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Participants v. Group Number (Female - Method 1)


Overall, Method 2 was shown to be a better method of organizing the divisions because

the standard deviation for the females had a decrease of 37.81% from Method 1to

Method 2 whereas the standard deviation for the males had an increase of 1.94% from

Method 1 to Method 2. For this reason Method 2 was proven to be a better strategy for

organizing the divisions in the open category.

R² = 0.9947

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Participants v. Group Number (Male - Method 2)

R² = 0.7481

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Participants v. Group Number (Female - Method 2)


Evaluating the Data:

Because the contestants start at different times, the total time taken by the contestants to

finish the triathlon could be modeled by adding 3 minutes multiplied by the quantity of

the wave number subtracted by one. By using this algorithm, the team found the total

time elapsed from the beginning of the triathlon to each contestant’s end of the triathlon.

Determining the Congestion

Scatter Plot:

The team used the time at which each person was at each stage during the triathlon to

measure congestion. To accomplish this measurement, the team used MS Excel to plot

the distribution of participants from the previous triathlon (Refer to Figure 1).

Box and Whisker Plot: The data was used to construct a Box and Whiskers plot in order

to illustrate the distribution of the contestants at different stages of the triathlon in terms

of quartiles (Refer to Figure 2).


Determining Advantages by Varying Distances

Equations

The team researched for studies done on how humans slow down over time when

performing aerobic exercises. (Riegel, 1981) One study found showed Equation A

(shown below) and the research done to help derive that equation.

𝑇2 = 𝑇1(𝐷2/𝐷1)1.06

In this equation, T2 is the time it takes for the contestant to run or bike for a distance of

D2 when given T1, the time it takes for the contestant to run for a distance of D1.

Applications of Equation A

Assuming that the distance of the swimming portion cannot be changed, the only two

sections of the triathlon that can be varied are the distances for running and biking. When

varying distances, the team established variables that remained constant. These variables

were D1, the given distance of each section in meters, and T1, the time it takes each

person to complete each section, in seconds. A control variable, D2, or the new altered

distance was also established. With this information, Equation A would yield the new

time, T2. T2 is the time for each person to travel D2.

Increasing Distance

When increasing distance, the D2 variable was set to be greater than the given D1

value. For the biking section, the given value for D1 was 40000 meters and for the

running section the given value for D1 is 10000 meters. To see what effect

increasing the distance would have, the D2 values were set at 50000 meters and

12000 meters respectively. Once these values and the given T1 were substituted

into Equation A, the values of T2 were greater than T1.


Decreasing Distance

When decreasing distance, the D2 variable was set to be less than the given D1

value. For the biking section, the given value for D1 was 40000 meters and for the

running section the given value for D1 is 10000 meters. To see what effect

increasing the distance would have, the D2 values were set at 30000 meters and

8000 meters respectively. Once these values and the given T1 were substituted

into Equation A, the values of T2 were less than T1.

ANALYSIS

Advantages of a Lower T2 Value

Using Equation A and chosen values for new distances D2, it is possible to determine the

time, T2, it will take a person to travel the new distance. If D2 is less than the given D1,

then the T2 will be less than the given T1. A T2 that is less than T1 represents the fact that

with D2, the athletes will take less time on that section than with D1. The less time the

athletes spend on each part of the triathlon, the sooner the local roads can be open to the

public. Additionally, the less time that it takes for the athletes to complete the course, the

sooner more people can exit the course, effectively decreasing any possible congestion on

the course.

Advantages of the Order of the Contestants:

● The roads will not be closed for more than five and a half hours.


● The congestion of people will not be a problem (further discussed in the Analysis of the

Plots section).

● The slower people will be given a headstart, and therefore more time to finish the

triathlon.

Analysis of the Plots:

Scatter Plot:

Since the data illustrating 3217 people was fairly spread out, the results indicate low

congestion during the race, and the congestion will be lower during the actual triathlon

because there is an estimated 2000 to 2100 attendants. The crowd density will be

proportional to the data found with the 3217 people.

Box and Whiskers:

To get a more in depth understanding of the density distributions of the contestants, the

team used a box and whiskers plot. The first quartile and the second quartile were the

most compact, showing that the road will be most congested time between 3:58:41 pm

and 04:10:03 pm. During that time, 25% of the total runners will be crossing the finish

line. Even though 25% of the runners will be crossing the finish line in this time, many

runners will be spread out throughout the area ranging from the beginning of the running

section to the finish line. This means that despite having 25% of the total participants in

one area of the triathlon, the runners will not all be in the same location. Additionally,

using the city standards for the city of Boston, there are five lanes available for runners to

use (bostoncompletestreets.org, 2013). The total length of these five lanes is 43’ (13.1064


meters) and the length of the running area is 10000 meters (bostoncompletestreets.org,

2013). According to a Harvard medical study, the average person is 1.6’ (0.48768 meters)

wide and 0.9792’ (0.2984 meters) in length (bostoncompletestreets.org, 2013). With

these measurements, a total of roughly 900636 people can fit in the running section of the

triathlon. Overall, the 25% of the runners in the race would take up 0.005% of the total

area of the running course, making it highly unlikely that congestion would occur.


Conclusion

If a person collapses or drowns the roads will be unable to reopen and the road will stay closed

for more than five and a half hours. In order to minimize the chances of these events occurring,

water stations and medics will be stationed around the paths to supply contestants with hydration

and safety, and lifeguards will be positioned around the lake and in motor boats to make sure

every swimmer is safe. If a runner runs slower than expected, then the road might need to be

closed for more than five and half hours. An advantage of the model is that the model accounts

for a possible extra 5% over the expected contestants, hence increasing the flexibility of the

model.


References

Admin. (2016). Triathlon swim starts: Wave vs. time trail (and others). Retrieved from

http://completetri.com/triathlon-swim-starts-wave-vs-time-trail-and-others/

Bostoncompletestreets.org. (2013). Minimum Widths for Roadway Lanes - Boston Complete

Streets. (n.d.). Retrieved November 11, 2016, from

http://bostoncompletestreets.org/pdf/2013/LaneWidths_Chart.pdf

Griggs, J. M. (2001). Human figure average measurements Retrieved from

https://www.fas.harvard.edu/~loebinfo/loebinfo/Proportions/humanfigure.html

Riegel, Peter S. (May–June 1981). "Athletic Records and Human Endurance". American

Scientist. 69: 285–290.

https://www.fas.harvard.edu/~loebinfo/loebinfo/Proportions/humanfigure.html

problem a paper: swim, bike, and run - wpiusers.wpi.edu/~ncng/images/himcm.pdf · team 6285 himcm...

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