© 2016 Scott Adamson

2017 Excellence in Mathematics Contest

Team Project

School Name:

Group Members:

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2

Reference Sheet

Formulas and Facts

You may need to use some of the following formulas and facts in working through this project. You may not

need to use every formula or each fact.

A bh 2 2C l w 2A r

Area of a rectangle Perimeter of a rectangle Area of a circle

2C r 1

2A bh 2 1

2 1

y ym

x x

Circumference of a circle Area of a triangle Slope

12 inches = 1 foot 5280 feet = 1 mile 3 feet = 1 yard

1 mile = 1.61 kilometers 2.54 centimeters ≈ 1 inch 100¢ = $1

1 kilogram ≈ 2.2 pounds 1 ton = 2000 pounds 1 gigabyte = 1000 megabytes

1 mile = 1609 meters 1 gallon ≈ 3.8 liters 1 square mile = 640 acres

1 sq. yd. = 9 sq. ft 1 cu. ft. of water ≈ 7.48 gallons 1 ml = 1 cu. cm.

2V r h V lwh

34

3V r

Volume of cylinder Volume of rectangular prism Volume of a sphere

Lateral SA = hr 2 a

acbbx

2

42

cos

sintan

Lateral surface area of cylinder Quadratic Formula

3

TEAM PROJECT

2017 Excellence in Mathematics Contest

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The Team Project is a group activity in which the students are presented an open ended, problem situation

relating to a specific theme. The team members are to solve the problems and write a narrative about the

theme which answers all the mathematical questions posed. Teams are graded on accuracy of

mathematical content, clarity of explanations, and creativity in their narrative.

Part 1 – Introduction by Ryan Mac , FORBES STAFF , staff writer covering technology and e-commerce

This story appears in the August 23, 2016 issue of Forbes.

It’s not hard to spot players of the most popular smartphone game of all time. They

have a peculiar way of carrying their devices in front of them with one hand, says John

Hanke, the technology whiz behind Pokémon GO, as we stroll along the waterfront of

San Diego’s Seaport Village the day before his appearance in front of 7,000 fans at

Comic-Con.

“They’re playing,” says the 49-year-old, nodding at a couple holding hands while their

eyes fixate on their phones. “That guy standing with the backpack. Those people sitting over there.”

Since its July launch Pokémon GO, a free “augmented reality” game by

Niantic Labs in which players capture virtual characters mapped to real-

world locations, has piled up superlatives. Apple said the game had more

downloads in its first week than any other app in history. One in ten

Americans plays Pokémon GO daily, according to App Annie, and

SurveyMonkey estimates that the game is hauling in as much as $6 million

a day from in-app purchases in the U.S. alone (the game is available in 37

countries). Beyond the numbers, Hillary Clinton has invoked the game on the campaign trail, Justin Bieber has gone

Poké-hunting in Central Park, and a reporter was publicly chided for playing it at a State Department briefing. It almost

didn’t happen. Just 12 months ago Hanke was an increasingly restless Google employee (he launched Google Earth,

among other things) and his company, Niantic, was an overlooked gaming skunkworks lost in the search giant. As

Google reorganized itself into Alphabet, Niantic looked likely to be rolled back into the company’s Android division or

simply shut down. But Google had the wisdom to let Hanke seek outside investors and spin the company out. That

paved the way for Hanke to approach Nintendo and the Pokémon Co., which oversees the brand’s intellectual property,

and make the smartest mobile-gaming deal of all time.

For Google, the arrangement has worked out in spades. Google owns just under 30% of Niantic, whose game might hit

$5 billion in annual revenue, according to a Macquarie Group analyst.

4

Part 2 – Is Pokémon Go Still Popular?

Pokémon Go started in the United States in the Summer of 2016. It was an instant hit! But how long did its popularity

last? The graph below (www.slashgear.com) shows the Worldwide Daily Active Users of Pokémon Go over a period

of time in the Summer of 2016. Use this graph to answer the questions that follow.

1. Describe the rate of growth of the daily active users of Pokémon Go for the first two weeks of data shown in this

graph. Express this rate as the additional number of daily active users added each day.

Answers may vary as estimated values are read from the graph. The number of daily active users grew from nearly 0

prior to July 5, 2016 to about 42,500,000 two weeks later on July 19, 21016. This represents an increase of about

3,035,714 additional daily active users each day.

𝟒𝟐, 𝟓𝟎𝟎, 𝟎𝟎 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔

𝟏𝟒 𝒅𝒂𝒚𝒔≈ 𝟑, 𝟎𝟑𝟓, 𝟕𝟏𝟒 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔 𝒑𝒆𝒓 𝒅𝒂𝒚

Student might report a percent increase. Assuming there is an initial 1 daily user on July 5, 2016, we can express the

daily percent increase as about 351% daily growth.

√𝟒𝟐, 𝟓𝟎𝟎, 𝟎𝟎𝟎𝟏𝟒

≈ 𝟑. 𝟓𝟏 𝒐𝒓 𝟐𝟓𝟏% 𝒅𝒂𝒊𝒍𝒚 𝒈𝒓𝒐𝒘𝒕𝒉

2. Assuming that the rate of growth found in #1 continued, how long would it take for the number of active daily users

to reach 200,000,000 users?

On July 19, 2016, there were about 42,500,000 daily users. To reach the 200,000,000 level, an additional 157,500,000

active daily users are needed. Assuming the constant rate of about 3,035,714 daily users per day increase, the number

of daily users will reach 200,000,000 after an additional 52 days (approximately). This is a total of about 66 days from

the beginning.

𝟏𝟓𝟕, 𝟓𝟎𝟎, 𝟎𝟎𝟎 𝒂𝒅𝒅𝒊𝒕𝒊𝒐𝒏𝒂𝒍 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔

𝟑, 𝟎𝟑𝟓, 𝟕𝟏𝟒 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔 𝒑𝒆𝒓 𝒅𝒂𝒚≈ 𝟓𝟏. 𝟖𝟖 𝒅𝒂𝒚𝒔

The date predicted is about September 9, 2016.

If students use the daily percent growth of 251%, the 200,000,000 level is reached after just 1.25 more day or on July

21, 2016.

𝟒𝟐, 𝟓𝟎𝟎, 𝟎𝟎𝟎 ∙ 𝟑. 𝟓𝟏𝒕 =200,000,000

𝟑. 𝟓𝟏𝒕 =𝟐𝟎𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎

𝟒𝟐, 𝟓𝟎𝟎, 𝟎𝟎𝟎

𝒕 ≈ 𝟏. 𝟐𝟑 𝒅𝒂𝒚𝒔

5

Part 2 – continued…

3. Describe the rate of growth of the daily active users of Pokémon Go for the last two weeks of data shown in this

graph. Express this rate as the additional number of daily active users added each day.

Answers may vary as estimated values are read from the graph. The number of daily active users decreased from

about 34,000,000 on August 2, 2016 to about 28,000,000 two weeks later on August 16, 21016. This represents a

decrease of about 6,000,000 daily active users each day.

𝟔, 𝟎𝟎𝟎, 𝟎𝟎𝟎 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔

𝟏𝟒 𝒅𝒂𝒚𝒔≈ 𝟒𝟐𝟖, 𝟓𝟕𝟏 𝒇𝒆𝒘𝒆𝒓 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔 𝒑𝒆𝒓 𝒅𝒂𝒚

Student might report a percent decrease. We can express the daily percent decrease as about 351% daily growth.

√𝟐𝟖

𝟑𝟒

𝟏𝟒≈ 𝟎. 𝟗𝟖𝟔𝟐 or a daily decrease by a factor of 1.38% of the previous day’s amount.

4. Assuming that the rate of growth found in #3 continued, how long would it take for the number of active daily users

to reach 0 users?

On August 16, 2016, there were about 28,000,000 daily users. Assuming the constant decrease of about 428,571

daily users per day, the number of daily users will reach 0 after an additional 65 days (approximately). This is a total

of about 79 days from the August 2 date.

𝟐𝟖, 𝟎𝟎𝟎, 𝟎𝟎𝟎 𝒅𝒆𝒄𝒓𝒆𝒂𝒆 𝒊𝒏 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔

𝟒𝟐𝟖, 𝟓𝟕𝟏 𝒅𝒂𝒊𝒍𝒚 𝒖𝒔𝒆𝒓𝒔 𝒑𝒆𝒓 𝒅𝒂𝒚≈ 𝟔𝟓. 𝟑𝟑 𝒅𝒂𝒚𝒔

The date predicted is about October 21, 2016.

If student use the daily decay factor of 98.62%, the 1 user level (since we can’t find when the level is 0) is reached

after 1234 more days or about 3.4 years!

𝟐𝟖, 𝟎𝟎𝟎, 𝟎𝟎𝟎 ∙ 𝟎. 𝟗𝟖𝟔𝟐𝒕 =1

𝟎. 𝟗𝟖𝟔𝟐𝒕 =𝟏

𝟐𝟖, 𝟎𝟎𝟎, 𝟎𝟎𝟎

𝒕 ≈ 𝟏𝟐𝟑𝟒 𝒅𝒂𝒚𝒔

6

Part 3 – Hatching Eggs

In the game of Pokémon Go, there are opportunities to hatch eggs. To hatch an egg, the user must put the egg into an

incubator and, depending on the egg, the user must walk 2, 5, or 10

kilometers.

1. Suppose you walk a casual 2.5 miles per hour. How long will it take

for you to hatch a 10-km egg?

10-km is equivalent to approximately 6.211 miles. At a speed of 2.5

miles per hours, it will take about 2.48 hours to hatch the egg.

2. To hatch eggs more efficiently, people have tried to launch the Pokémon Go app while driving (or as a passenger) in

a car. However, there is a 6.52 MPH speed limit set within the game. Suppose that you were running a marathon

(26.2 miles) at a constant speed of 6.52 MPH and had your Pokémon Go app running. How many 2-km eggs could

be hatched? How many 5-km eggs? How many 10-km eggs? How long would it take for each?

First some conversions: 6.52 MPH is equivalent to about 10.5 km/hr and 26.2 miles is equivalent to 42.16 km.

Therefore, the following number of eggs could be hatched while running 42.16 km:

2-km eggs: 𝟒𝟐.𝟏𝟔 𝒌𝒎

𝟐 𝒌𝒎 𝒑𝒆𝒓 𝒆𝒈𝒈≈ 𝟐𝟏 𝒆𝒈𝒈𝒔

𝟐 𝒌𝒎

𝟏𝟎.𝟓 𝒌𝒎/𝒉𝒓≈ 𝟎. 𝟏𝟗 𝒉𝒐𝒖𝒓𝒔

5-km eggs: 𝟒𝟐.𝟏𝟔 𝒌𝒎

𝟓 𝒌𝒎 𝒑𝒆𝒓 𝒆𝒈𝒈≈ 𝟖 𝒆𝒈𝒈𝒔

𝟓 𝒌𝒎

𝟏𝟎.𝟓 𝒌𝒎/𝒉𝒓≈ 𝟎. 𝟒𝟖𝒉𝒐𝒖𝒓𝒔

10-km eggs: 𝟒𝟐.𝟏𝟔 𝒌𝒎

𝟏𝟎 𝒌𝒎 𝒑𝒆𝒓 𝒆𝒈𝒈≈ 𝟒 𝒆𝒈𝒈𝒔

𝟏𝟎 𝒌𝒎

𝟏𝟎.𝟓 𝒌𝒎/𝒉𝒓≈ 𝟎. 𝟗𝟓 𝒉𝒐𝒖𝒓𝒔

A total of 21 two kilometer eggs, 8 five kilometer eggs and 4 ten kilometer eggs could be hatched during the running of

the marathon. At an average, constant speed of 10.5 km/hr, the entire marathon would be completed in about 3.92

hours. Every 0.19 hours (11.4 minutes), a 2-km egg would be hatched. Every 048 hours (28.8 minutes), a 5-km egg

would be hatched. Every 0.95 hours (57 minutes) a 10-km egg would be hatched.

7

Part 3 – continued…

Some people have had the idea of attaching their phone (with the Pokémon Go app

running) to a ceiling fan to hatch their eggs while relaxing at home or avoiding unpleasant

weather outside. Suppose that the Pokémon Go app can measure the precise distance

travelled while spinning on the fan.

3. If the ceiling fan has 24-inch blades and has a high speed of 289 revolutions per minute,

how long will it take to hatch a 2-km egg?

A 24-inch blade (assuming phone is taped right on the end) will sweep out a

circumference of approximately

𝑪 = 𝟐𝝅 ∙ 𝟐𝟒 ∙≈ 𝟏𝟓𝟎. 𝟖 𝒊𝒏𝒄𝒉𝒆𝒔 in one revolution.

Also, 2-km is equivalent to about 1.24 miles, 6,561.68 feet, and 78,740.16 inches. This is 𝟕𝟖,𝟕𝟒𝟎.𝟏𝟔 𝒊𝒏𝒄𝒉𝒆𝒔

𝟏𝟓𝟎.𝟖 𝒊𝒏𝒄𝒉𝒆𝒔 𝒑𝒆𝒓 𝒓𝒆𝒗𝒐𝒍𝒖𝒕𝒊𝒐𝒏≈ 𝟓𝟐𝟐. 𝟏𝟓 revolutions of the fan.

Therefore, it will take just 𝟓𝟐𝟐.𝟏𝟓 𝒓𝒆𝒗𝒐𝒍𝒖𝒕𝒊𝒐𝒏𝒔

𝟐𝟖𝟗 𝒓𝒆𝒗𝒐𝒍𝒖𝒕𝒊𝒐𝒏𝒔 𝒑𝒆𝒓 𝒎𝒊𝒏𝒖𝒕𝒆𝟏. 𝟖𝟐 𝒎𝒊𝒏𝒖𝒕𝒆𝒔 to hatch a 2-km egg. However, it should be

noted that the GPS system will likely not recognize this circular distance since the phone is not traveling away from

the home.

8

Part 4 - Playing the Game Efficiently

Suppose you have two 10-km eggs, three 5-km eggs, and four 2-km eggs that you would like to hatch. You also have

one incubator that you can use as many times as you like and an incubator that you can only use 3 times. What is the

least number of kilometers you would need to travel to hatch all the eggs? Clearly describe your strategy so that it

convinces the reader that it is the most efficient (in terms of distance traveled) way to hatch all the eggs. Note that an

incubator can hold only 1 egg at a time.

Best solution:

Put at 10-km egg in the 3 use incubator. While this 10-km is hatching, you can simultatneously

hatch two 5-km eggs (one at a time) in the infinite use incubator – total distance walked is 10-km

so far.

Put the other 10-km egg in the 3 use incubator. At the same time, a 5-km and two 2-km eggs can

be hatched. Also, another 2-km egg will be half-way completed – total distance walked is 10-km

plus the previous 10-km for a total of 20-km so far.

Put the last 2-km egg in the 3 use incubator. While it hatches, the other half of the 2-km egg from

step 2 also hatches – total distance walked is 2-km plus the previous 20-km for a total of 22-km.

10-km 10-km

5-km 5-km 5-km

2-km 2-km 2-km 2-km

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Part 4 – continues…

10

Part 5 – Pokémon Probabilities

There are many reasons why players of Pokémon Go benefit from hatching eggs. Hatching eggs helps the Pokémon

player obtain Stardust and Candy and gives them a greater chance of obtaining rare Pokémon. However, the player

does not know what Pokémon they will hatch! The charts below show the probability of hatching particular Pokémon.

Use the charts to answer the questions that follow.

11

Part 5 – continued…

1. Suppose that over time, you collected fifty 2-km eggs for your Pokémon game. How many

of these eggs would you expect hatch the Pokémon Squirtle?

According to the chart, 4.8% of 2-km eggs will hatch Squirtle. Since, 𝟎. 𝟎𝟒𝟖 ∙ 𝟓𝟎 = 𝟐. 𝟒,

we would expect about 2 Squirtle to hatch from our eggs.

2. Suppose you have two 5-km Pokémon eggs to hatch. What is the probability that you hatch both Voltorb and

Tangela?

The probability of obtaining both Voltorb (2%) and a Tangela (1%) is 𝟐

𝟏𝟎𝟎∙

𝟏

𝟏𝟎𝟎=

𝟐

𝟏𝟎,𝟎𝟎𝟎 or 0.02%.

3. Suppose you have two 10-km Pokémon eggs to hatch. What is the probability that you hatch either Eevee or

Magmar?

𝑷(𝑬𝒆𝒗𝒆𝒆 𝒏𝒐𝒕 𝑬𝒆𝒗𝒆𝒆) = (𝟎. 𝟏𝟕𝟕)(𝟎. 𝟖𝟐𝟑) = 𝟎. 𝟏𝟒𝟓𝟔𝟕𝟏 or about 14.6%

𝑷(𝑴𝒂𝒈𝒎𝒂𝒓 𝒏𝒐𝒕 𝑴𝒂𝒈𝒎𝒂𝒓) = (𝟎. 𝟎𝟖𝟖)(𝟎. 𝟗𝟏𝟐) = 𝟎. 𝟎𝟖𝟎𝟐𝟓𝟔 or

about 8.0%

Therefore, the probability of hatching either Eevee or Magmar is

𝑷(𝑬𝒆𝒗𝒆𝒆 𝒏𝒐𝒕 𝑬𝒆𝒗𝒆𝒆) + 𝑷(𝑴𝒂𝒈𝒎𝒂𝒓 𝒏𝒐𝒕 𝑴𝒂𝒈𝒎𝒂𝒓) ≈ 𝟐𝟐. 𝟔%