2018 Excellence in Mathematics Contest Team Project Level II (below Precalculus) School Name: Group Members:

Transcription

1 2018 Excellence in Mathematics Contest Team Project Level II (below Precalculus) School Name: Group Members:

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 C 2l 2w 2 A r Area of a rectangle Perimeter of a rectangle Area of a circle C 2 r 1 A bh billion acres Circumference of a circle Area of a triangle Habitable land on Earth a b c 5280 feet = 1 mile d = rt Pythagorean Theorem 1 gallon = cu. ft centimeters = 1 inch 2 h 4.9t v0t h0 2 h 16t v0t h0 1 kilogram = 2.2 pounds 1 meter = inches 1 gigabyte = 1000 megabytes 1 mile = 1609 meters 1 gallon = 128 ounces 1 square mile = 640 acres 1 sq. yd. = 9 sq. ft m y ml = 1 cu. cm. x 2 y x V r h V Area of Base height V r 3 Volume of cylinder Volume Volume of a sphere Lateral SA = 2 r h Lateral surface area of cylinder b b 2 4ac x 2a Quadratic Formula tan sin cos Ideas contained in this Team Competition come from: How Many Licks? Or, How to Estimate Damn Near Anything, by Aaron Santos, Ph.D., Running Press,

3 TEAM PROJECT Level II 2018 Excellence in Mathematics Contest 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: Background In this Team Project, you may find yourself wondering, How in the world will we ever respond? That s the point to have you make assumptions, predictions, estimations, and to fully communicate a response! This project is all about estimating being approximate but not just randomly. Your work will be judged on how well you articulate your work, how well you convince the judges that you have made reasonable assumptions, and how well you justify your approximations with sound mathematical argumentation. Consider this excerpt from the book, How Many Licks? Or, How to Estimate Damn Near Anything, by Aaron Santos, Ph.D. Approximation works as an idea filter that can be used to weed out bad ideas when making a decision. Whether you re a businessman deciding which product to pursue, a congressman voting on whether or not to build a fence around the Mexican border, or a physicist trying to detect the Higgs Boson, approximation is the first technique you should use as a feasibility test. Let s say you re a government official working on a missile defense plan. Do you pursue it immediately, potentially wasting billions of taxpayer dollars only later to find out whether or not it ever had a chance of stopping a nuclear attack? Or should you first make a rough estimate of the plan s likelihood of success, potentially saving time and money while you come up with a legitimate plan? If you estimate the plan has a 90% chance of success, you definitely pursue it. Maybe even if you estimate it has only a 10% chance, you still pursue, it. But, if your estimate concludes its chances of realization are less than the chance of winning the lottery, it would be foolish to ever consider it. Even without an advanced engineering degree, estimations like these are possible and necessary for making some of the critical decisions we face every day. Whether it be missile defense or simply when to cross the street, we make estimations all the time, and having a sound mathematical footing only improves accuracy. In addition to providing an idea filter, getting in the habit of formally approximating things also boosts one s numerical prowess, specifically one s ability to conceptualize very big (and very small) numbers. Although you may not yet have an appreciation for the difference between a billion and a trillion, experience using these numbers builds an understanding of them quickly. One rapidly develops numerical landmarks that act as a conceptual guide. For example, at the time of this writing, a billion is about one-seventh the world population and a trillion is about one-eleventh the national debt. With simple arithmetic and a little bit of practice, you can estimate just about anything (no matter how large or small) and grow comfortable understanding big numbers in the process. So, get to it! For each of the situations in this project, your task is to prepare a well-articulated explanation of your response. At first, you may wonder how you will ever respond but know that in each case, with appropriate assumptions that you will need to make and clearly state, you will be able to respond! Work together, be creative, use the resources provided with each situation (some are helpful, some not so helpful), and do your best to create a well-crafted, clear, mathematically sound justification for your response. And, just to say it one more time clearly state your assumptions and estimations that lead to your final response! Have fun! 3

4 Part 2: Agony of Da Feet It has been said that you have to walk a mile in a man s shoes before you can judge him. How many miles does a person walk in a lifetime? Santos, p. 26 Sample Solution judge student work based on their assumptions, clarity, justifications, accuracy, and reasonableness. Assume: 5 80 years of life hours We walk more than 0.5% of the time but less than 50% of the time assume 5% of time A person walks 1.0 MPH 5 total hours waking hours walking hours lifetime total hours waking hours mile , 000 miles per lifetime hour 4

5 Part 2: Resources Top 101 Countries by Life Expectancy 5

6 Part 2: Resources Healthy Walking Habits

7 Part 3: Walking on the Sun Everyone s had awful vacations: children complaining, Dad obsessing taking pictures, Mom lecturing on how getting there is half the fun, and of course, the hours and hours of being stuck in a packed car all for a few hours on a humid beach in the baking sun. But if you think the blistering drive to Florida took forever, just imagine how long a trip to the actual Sun would be. Santos, p. 51 How long would it take to drive to the sun? Sample Solution judge student work based on their assumptions, clarity, justifications, accuracy, and reasonableness. Assume: 7 Distance to the Sun miles Average driving speed 60 MPH 7 miles miles 60 1,550, 000 hours 177 years hour 7

8 Part 3: Resources 8

9 Part 4: Eat It, Spider-Man! When it comes to superhero movies, one must always maintain some suspension of disbelief. For instance, in the Spider-Man series, it s perfectly natural for audiences to believe that getting bitten by a radioactive spider gives you super powers, but are we really expected to believe all of that web came out of one body? Isn t there some physical law that says matter can t be created on the fly? Santos, p. 83 How many pounds of food does Peter Parker have to eat to produce an equivalent amount of web each day? Sample Solution judge student work based on their assumptions, clarity, justifications, accuracy, and reasonableness. Assume: In the movies, webs can be 20 meters or about 66 feet long At 20 meters or 66 feet, Spidey would have to shoot 80 webs to travel a mile Suppose he travels 1 mile each day by web slinging Suppose that the web is 1.0 cm thick Density of spider web silk is about 1.3 grams per cubic cm. The web cord is cylindrical cm grams 80 webs 0.5 cm ,363 grams 360 pounds web cm 2 3 9

10 Part 4: Resources screenrant.com en.wikipedia.org/wiki/spider_silk A dragline silk's tensile strength is comparable to that of high-grade alloy steel ( MPa), [11][12] and about half as strong as aramid filaments, such as Twaron or Kevlar (3000 MPa). [13] In 2018, a wood-based nanofiber achieved tensile stiffness eight times greater and with higher tensile strength than spider silk. [14] Consisting of mainly protein, silks are about a sixth of the density of steel (1.3 g/cm 3 ). As a result, a strand long enough to circle the Earth would weigh less than 500 grams (18 oz). (Spider dragline silk has a tensile strength of roughly 1.3 GPa. The tensile strength listed for steel might be slightly higher e.g GPa, [15][16] but spider silk is a much less dense material, so that a given weight of spider silk is five times as strong as the same weight of steel.) The energy density of dragline spider silk is roughly J/m 3. [17] Silks are also extremely ductile, with some able to stretch up to five times their relaxed length without breaking. 10

11 Part 5: Lincoln s Penny The president is a very busy man. As leader of the free world, former President George Bush had numerous responsibilities from balancing the budget to protecting our American citizens to spending 20 percent of his days in office vacationing at his Crawford ranch. One would imagine that he should be adequately compensated. If he were a king, he could collect his weight in gold, but perhaps instead, we could just fill the Oval Office with pennies. Santos, p. 73 What s worth more money: the President s actual salary or the number of pennies that could fit in his office? Sample Solution judge student work based on their assumptions, clarity, justifications, accuracy, and reasonableness. Assume: The Oval Office measures 11 m on the long axis, 8.8 meters on the short axis and 5.6 meters in height 3 If it were the Box Office, the total volume would be 11 m 8.8 m 5.6 m 540 m 2 3 The volume of a penny is about (1 cm) (0.15 cm) 0.47 cm Presidential salary is $400,000 per year m 540 m billion pennies penny About $11 million worth of pennies can fit in the Oval Office. That s about 30 times what the President makes. 11

12 Part 5: Resources President Donald Trump plans to donate his presidential salary to charity at the end of the year. "The way that we can avoid scrutiny is to let the press corps decide where it should go," White House press secretary Sean Spicer told reporters Monday. During a question-and-answer session on Twitter in 2015, Trump said he would give up his presidential salary if he were elected. "I won't even take one dollar," Trump said. Turns out, Trump does have to take at least one dollar, something he acknowledged after the election. George Washington didn't want to take a salary, either. But the framers of the Constitution thought the president needed a fixed compensation to remain truly separate from Congress, so they wrote it into the Constitution. "The legislature, with a discretionary power over the salary and emoluments of the Chief Magistrate, could render him as obsequious to their will as they might think proper to make him. They might, in most cases, either reduce him by famine, or tempt him by largesses, to surrender at discretion his judgment to their inclinations," Alexander Hamilton explained in "Federalist No. 73." So in 1789, Washington got paid $25,000 a year. Congress has increased that salary five times since; the latest raised George W. Bush's compensation to $400,000 in Trump won't be the first president to donate his earnings. Both John F. Kennedy and Herbert Hoover did. Hoover also used some of his salary to boost his staff's paycheck. 12

13 Part 6: Cartoon-y Images My filmmaker friend occasionally asks my advice on how to build different things for his movies. Though I like the attention, I m usually somewhat perplexed because my background is in physics, not engineering. (It s almost as bad as when people ask me to program their VCR clocks.) But he has one idea that I really enjoyed: He wanted to know Santos, p. 171 How many helium balloons are needed to carry a man down the street if he s hovering just a few feet off the ground? Sample Solution judge student work based on their assumptions, clarity, justifications, accuracy, and reasonableness. Assume: Density of Helium is 0.18 kg/m 3 and the density of air is 1.25 kg/m 3 Each balloon is a sphere with a 10 cm radius. The volume then is V 4 (10 cm) 4200 cm 3 Mass of a person is 65 kg (~140 pounds) The balloons will have to overcome the mass of the person. 65 kg 3 kg kg m m m balloon balloons 3 3 You would need about 14,000 balloons! 13

14 Part 6: Resources Balloon Lift with Lighter than Air Gases It has long been known that if immersed in a gas or liquid, an object will displace a volume of that gas or liquid equivalent to its own volume. By comparing the weight of the object vs the weight of this displaced volume of gas or liquid, you can determine if the object will float or sink like the proverbial stone. Some dude allegedly thought this up while lounging in his bath. When a balloon is filled with something other than air and then released in air, it will float or sink based on the same principle. If the weight of the volume of air displaced by the balloon is less than the weight of the balloon and the gas inside, the balloon will drop to the ground. If the weight of the air displaced by the balloon is greater than the weight of the balloon and the gas inside, the balloon will float upwards. This force, or buoyancy, either positive or negative, is exactly the difference in the weight of the balloon and its contents, versus the weight of the volume of air displaced. As volumes get large on any balloon, the problem of distributing the load stress becomes great. Large plastic lifting balloons have extremely strong strips incorporated into the multiple seams that join the gores together running from top to bottom. Designs from the 1800's used nets to contain the balloon and distribute the stress of the suspended load. One of the most famous of the Civil War balloons was made by the Confederacy of donated undergarments of the ladies of the South, all silk and the only source of silk the South had at the time. It was hoped that it would counter the shocking success of Union balloons which were providing airborne intelligence of opposing troop movements that was proving devastating. The southern patchwork balloon only flew once before it was captured by the Union, a disappointment and an afront for which General Longstreet claimed he never found it in his heart to forgive the Union. 14

15 Part 7: Into the Wolf s Mouth Fairytales are rife with hyperbole and overstatement. Consider the Big Bad Wolf chasing the Three Little Pigs. Santos, p. 135 How many wolves would it take to actually huff and puff and blow the house down? Sample Solution judge student work based on their assumptions, clarity, justifications, accuracy, and reasonableness. Assume: A category 5 hurricane has 70 m/s winds (~155 MPH). The same flow rate or wind speed is needed to blow the house down. Suppose a three-story house is 10 m tall and just as wide. The area of one side of the house is 100 m 2 (~1,100 sq. ft.). Flow rate is calculated by multiplying the speed of the winds by the cross-sectional area of the house. 2 3 Based on our assumptions, the flow rate is 70 m/s 100 m 7000 m /s. Suppose wolves can huff and puff and blow the house down at the same speed they sneeze and that they sneeze at the same speed as a human which is 45 m/s (100 MPH). The opening of the wolf s mouth is about 3 cm by 3 cm or 9 cm 2. 3 m 7000 s m s 4 3 m wolf wolves It would take about 170,000 wolves to blow a house down! 15

16 Part 7: Resources