A Final Word

I'm closing up this math applications blog for now. I may come back to it at some point. I was writing this as a tie-in to my book. However, I've got another job now. Its a long story. I have had fun doing this, but its time has come. The past postings still exist, of course, and you can check out any of them for ideas regarding applications of mathematics. Even if a particular one is not exactly what you were looking for, it might cause you to think along the lines of something else that better suits your purpose.

And here is something kind of interesting. This happens to be my 99th posting. That made me think that maybe I needed to do a 100th. No, I'll think I'll leave it as it is because 99 represents the incompleteness that... OK, I've got nothing, but there is probably something profound in ending on 99. Maybe I'll figure that out and make it my 100th post some day.

Absolute Value

Absolute value is a tricky thing. Students love it because it is so easy. They probably come away thinking, "Could I have done this right? That just felt way to easy. And even if I did do it right, it seams pretty pointless." There actually are mathematics applications to absolute value. Here are a few.

Average Deviation – There are a number of formulas that measure the variability of data. A common one is the standard deviation. However, average deviation is similar and easier to compute. The average deviation simply finds the average distance each number is from the mean. To find the average deviation, the distance from the mean is found for each piece of data in the set. Those distances are added and then divided by the number of pieces of data. If the mean is 32, we would want 28 and 36 to both be considered positive 4 units away from the mean. Absolute value is used so there are no negative values for those distances.

Example: 

- A set of data is {21, 28, 31, 34, 46}. The mean average is 32.The average deviation is 6.4.

Statistical Margin of Error – As mandated by the U.S. Constitution, every ten years the government is required to take a census counting every person in the United States. It is a huge undertaking and involves months of work. So how are national television ratings, movie box office results, and unemployment rates figured so quickly – often weekly or even daily? Most national statistics are based on collecting data from a sample. Many statistics that are said to be national in scope are actually data taken from a sample of a few thousand. Any statistic that is part of a sample is subject to a margin of error. (In 1998, President Clinton attempted to incorporate sampling in conducting the 2000 census, but this was ruled as unconstitutional.)

Example:

- On October 3, 2014 the government released its unemployment numbers for the month. Overall unemployment was listed at 5.9%. The report also stated that the margin of error was 0.2%. Government typically uses a level of confidence of 90%. Thus there is a 90% chance that the actual unemployment rate for the month was x, where |x-5.9| ≤ 0.2.

Richter Scale Error – The Richter scale is used to measure the intensity of an earthquake. However, like many measurements, there is a margin of error that needs to be considered. Scientists figure that the actual magnitude of an earthquake is likely 0.3 units above or below the reported value. If an earthquake is reported to have a magnitude of x, the difference between that and its actual magnitude, y, can be expressed using absolute value:  |x-y| ≤ 0.3.

Body Temperature – “Normal” body temperature is assumed to be 98.6° F. For any student that has made the case that anything other than 98.6° prevents their attendance at school, there is good news. There is a range surrounding that 98.6 value that is still considered in the normal range and will allow your attendance at school. Your 99.1° temperature is probably just fine. Supposing plus or minus one degree is safe, an expression could be written |x-98.6| ≤ 1.0, which would represent the safe range. Why is the absolute value a necessary part of this inequality? Without it, a temperature of 50 degrees would be considered within the normal range, since 50-98.6 = -48.6, which is, in fact, well less than 1.0.

Ten Cool Things about Laplace

I thought it was time to look at another interesting mathematician. This week it is Pierre-Simon LaPlace.

1. Lived from 1749 to 1827, all in France.
2. Married at age 39 to an 18 year old.
3. Made important contributions to the method of least squares - used to find a best fitting line.
4. Wrote the five volume Celestial Mechanics contributing greatly to a theory of the origins of the universe.
5. Appointed by Napoleon Bonaparte to be Minister of the Interior of France.
6. Later regretting this, Napoleon later stated, "Laplace was not long in showing himself a worse than average administrator."
7. Very possible apocryphal, but Napoleon was speaking to Laplace on the influence of God on a a particular situation to which he replied, "I had no need of that hypothesis."
8. Commenting on this story, Stephen Hawking said, "I don't think that Laplace was claiming that God does not exist. It's just that he doesn't intervene, to break the laws of science."
9. When he died, his brain was removed and displayed.
10. He is buried in Paris in the Pere Lachaise Cemetery along side other star-studded famous residents Balzac, Sarah Bernhardt, Bizet, Maria Callas, Chopin, Joseph Fourier, Yves Montand, Jim Morrison, Marcel Proust, Rossini, and Oscar Wilde.

Sermon Stats

In sermon notes in a church bulletin it stated, "The probability Jesus could have fulfilled even eight of these prophesies is 1 in 10 to the 17th power (1 in 100,000,000,000,000,000)". This was a statistic taken from a book, although I don't know the title. I thought there is a math application in there somewhere.

I thought that small a number might be almost incomprehensible to most. Maybe to everyone. It

reminds my of something David Letterman said once regarding buying a lottery ticket. A particular lottery was at a near record amount and lots of people were buying them. He wanted people to consider that if you buy a ticket, your chance of winning is only slightly more than if you don't buy one. Incidentally, I was in the audience for one of his shows during his final month. Hilarious. I am including a picture for no other reason than I love Dave. Back to math.

I considered a couple of ways to tie this probability to other situations. How does this probability compare with chances in rolling a die? In flipping a coin?

Well, the chances of rolling a "6" are one in six. How many consecutive rolls would correspond to the above probability?

1 / 10¹⁷ = 1 / 6^x
10¹⁷ = 6^x
Taking the common log of each side, we get:
17 = x(log6)
x = 21.85

So, at least 21 consecutive rolls coming of 6.

Similarly with flipping the coin. The coin has only two outcomes, so:

1 / 10¹⁷ = 1/2^x
After a few steps we get x = 56.47

56 heads in a row. Unlikely.

If worried about church vs state issues, a teacher could come up with other kinds of problems. The actually probablility of winning a certain lottery, winning the grand prize in the McDonald's Monopoly Game. For example, I just looked up on-line that the probability of getting the Boardwalk piece - 1 in 602,000,000.

Good Luck.

Evaporation

When I was a lad, I remember looking a drops of rain that had plopped on the sidewalk. It was a light rain so I could make out the individual drops. They gradually evaporated. I noticed that if I used my finger and spread the raindrops, out they evaporated faster. At my tender age I had no idea why. Still not 100% certain, but my guess now is that if you have a drop of water it is losing molecules, i.e., evaporating, from its surface. If you take a drop of water, it is evaporating at a certain rate. If you separate that drop into two drops, it will evaporate faster because there is much more surface area for which it can use to evaporate.

So, let's examine this as a math application. Suppose a drop of water is spherical and has a volume of 10 whatevers. Then suppose we separate that drop into two drops of volume 5 whatevers each. Then we look at their total surface areas. Will they come out the same?

We start with the fact that it has a volume of 10:  (4/3)πr³ = 10
If we solve for r we get a value of r = 1.3365
We can then find its surface area: 4π(1.3365)² = 22.446

Now what if we now have two spheres of 5 each.
Their radii would be:  (4/3)πr³ = 5, so r = 1.0608
The surface area of one drop is 14.140. There are two drops, though, so the total surface area of them would be 28.28

Comparing the two situations, we can in fact state that, since 28.28/22.466 = 1.26, there is 26% more surface area, and so I will postulate a 26% faster drying time for the two drops over the one drop.

That was to be the end of the story, but I thought what if you separate one spherical drop into two? Is it always going to be a 26% greater surface area.

Unfortunately this is beyond my skill level to show all this. You might recall that I only recently found how to write integer exponents. This process involves having the cube root of a fraction all taken to a power of two and other assorted difficulties. So let me map this out leaving a few gaps for you or students to work through. It really is a great problem, though, with the opportunity to review simplifying - some major simplifying.

• Let us say that  (4/3)πr³ = V
• Solve for r
• Substitute this expression into 4πr²
• Now, find the radius for a sphere that his half the original volume:    (4/3)πr³ = (v/2)
• Substitute this r into 4πr²
• Make a ratio of the two radii and simplify 
• You end up with cube root of 16 divided by 2, which is 1.26
• Ta-da. An increase of 26%

Satisfying

Predicting Win Percentages

Continuing on from last weeks post regarding the website fivethirtyeight.com and how they come up with their information. Last week was about how they look at various political polls and how they rank them. I found that quite interesting.

Even more interesting to me is how they come up with in-game percentages as to who is going to win. We all do that to some degree. Five minutes to go and your team has a ten point lead. You are probably going to win. So it is over 50%. But is it 60%? 85%? They know. At least I would say that their guess is as good as it gets.

I want to give Jay Boice and Nate Silver credit because I'm just relaying what they say is how their group comes up with those percentage win chances. I will try to do their explanation justice. So with a bit of paraphrasing, here we go,

• You you are ahead by 10 a with five minutes to go. The question becomes - How often have teams in that same situation done that in the past?
• They use regression analysis based on various game situations in the past. "The past" being the scores from all of the NCAA games over the past five years. 
• It makes a difference if that team that is ahead is really the better team, so they also factor in the pre-game win probabilities. That team currently in the lead may be more lucky than good.
• Finally, what is the current situation? It's five minutes to go. But who has the ball. Is one of the teams getting ready to shoot free throws?
• They don't account for everything, e.g., a player has fouled out and won't be available the rest of the game. That certainly could have an impact. 
• There probably are a number of factors that are just too much to deal with, so they don't.

Their results are pretty impressive. I haven't checked them out in real time. Its always after a game has been played. I'll have to remember to do that. Looking at them after the fact, though, their results seem pretty impressive. You can see some of their March Madness work here: 2017 March Madness Predictions

Rating the Polls

I was going to call this week's post "March Mathness" and talk a little about the NCAA tournament. Let's do that next week. Let me go ahead, though, and apologize for the title now. I'm sure I'm not the only one to use this type of play on "March Madness". That still doesn't make it right.

There is a nice website by the name of fivethirtyeight.com. It presents information regarding polls and polling data (The 538 part comes from the fact that there are 538 electors in the electoral college.) One interesting part of the website is looking at various polls (there are a lot more than I would have imagined - they rate over three hundred polling firms).

The reason I got there is because I was trying to figure out how their site, can come up with in-game information like Arizona is ahead of St. Johns 55 to 46 with 3:38 left to play, thus Arizona has an 89% chance of winning. Wow. It's clear Arizona would probably win, but how do they come up with a percent like that? Anyway, we'll look at that next week.

I got side-tracked with a section that speaks to how they rate various polls. For example the Trump/Clinton election did not come out as most had predicted. Some polls are better than others. They rate them all. For example, one of the best seems to be the ABC News/Washington Post poll. On the other had, an organization called Research 2000 is not. An overview of their methodology is at:

https://fivethirtyeight.com/features/how-fivethirtyeight-calculates-pollster-ratings/

They don't really give enough information to show exactly how they do it. That would probably be beyond me anyway. Let me tell you something they have used in the past. It is an especially cool math application since it has a square root stuck in there.

Total Error = Square Root of (Sampling Error + Temporal Error + Pollster Induced Error)

Why don't polls come out exactly right:

1. Sampling Error:  Sampling not enough people or not getting a representative sample
2. Temporal Error:  The farther away it time a poll is from the event; the more error
3. Pollster Induced Error:  Seems to be kind of a catch-all category for other things that can go wrong, such as assuming a too high or too low voter turnout.

Something else interesting they talk about is the concept of "herding". The companies that do the polling want to look good. It does not look good if they've wandered too fall away from the rest of the herd. If most every other poll has candidate A having around 55% of the vote and you predict he'll have 73%, you might make an "adjustment" to your results. Or you simply chose to not publish those results in which your company seems to be way off. 

That and other factors make it pretty complicated. Polling itself is complicated and then ranking the pollster even more so. 

I hope I've done justice to what they do. If you read what they have to see on their website you can see the complexity involved.

Next week, March Madness. Don't worry it will still be going on. In fact, it is actually March and slopping over into a little bit if April Madness. 

Top Ten Applications of Pi

It is pi day!!! It snuck up on me. I was going to make a top ten of cool facts about pi - by stealing them from the internet, of course. But most of those are not math applications, just cool trivia with virtually no applications, e.g., in a Star Trek episode, Spoke once defeated a computer by commanding it to compute the last digit of pi.

Brainstorm. Combine the two into the top ten applications of pi. And also doing so by stealing them from the internet. All wrapped up in one tidy package. By the way, I understand some of these applications fully and some not very well at all. I will try to give a short internet-stolen explanation for these.

1. Circumference of a circle.
2. Area of a circle.
3. Volume of a sphere.
4. Surface area of a sphere.
5. Cosmological constant - Value of the energy density of the vacuum of space. Originally introduced by Albert Einstein in 1917.
6. Heisenberg's uncertainty principle - Precision with which certain pairs of physical properties of a particle (such as momentum and position) can be known. Joke break: Heisenberg is driving down the road and is pulled over by a police officer. Police: "Do you have any idea how fast you were going?" Heisenberg: "No, but I know exactly where I am."
7. Einstein's field equations of general relativity - Describes the fundamental interaction of gravitation as a result of space time being curved by mass and  energy.
8. Coulomb's Law - Force between two charged particles.
9. Period of a pendulum.
10. Buckling formula - Finds the stress an object can handle before "buckling" would occur.

Hidden Figures

This is kind of a review of the movie/book Hidden Figures. Before seeing either one, I was a little concerned that it was going to be preachy. We need to treat African American people / women with respect. Hollywood, you don't need to tell us that. The people that already know that don't need to be told. And the people that don't know that probably aren't going to pay money to see this movie anyway. But it wasn't like that - the movie or the book. It just told their story.

If you want the true story, books are usually a better bet than the movie. The movie seemed quite in line with the book, but there were a few things. The two hour movie obviously had to leave a lot of stuff out that was in the book. There was also a time one of the ladies was at a chalkboard. She was impressing the room with the math she was doing. Some of it was a little unrealistic. Most people can't rattle off sin(23) to ten-thousandths place from memory - stuff like that. But, I'm quibbling. The movie was really good.

I got the book. And to be honest, I did some skimming in parts. I may get around to reading every word at some point, but that will be a ways off. There were a couple of cool things dealing with math applications that I thought was interesting. First of all, the book is Hidden Figures and is written by Margo Lee Shetterly. There. I hope that covers me from violating any copy write issues. Regardless, here we go.

Where do systems of equations take place in real life? Well check this out: "Modeling flight at transonic speeds was a particularly knotty problem, because of the subsonic and supersonic winds that passed over the plane or model simultaneously. Aerodynamic equations describing transonic airflows might contain as many as thirty-five variables. Because each point in the airflow was dependent on the others, an error made in one part of the series would cause an error in all the others. Calculating the pressure distribution over a particular airfoil at a transonic speed could easily take a month to complete for the most experienced of mathematicians." (pages 137, 138)

I once had the father of a student tell me about an older child of his that was in the NASA astronaut program. He said that their astronauts had to have a very good mathematics background in case something happened to the on-board computers. Another part of this book spoke to that issue:  "An astronaut stranded hundreds of thousands of miles from Earth is like a mariner from a previous age, adrift in the most remote part of the ocean. So what do you do when the computers go out? This was precisely the question Katherine [Johnson] and her colleague Al Hamer had asked in the late 1960s, during the most intense preparations for the first Moon landing. And in 1967, Johnson and Hamer coauthored the first of a series of of reports describing a method for using visible stars to navigate a course without a guidance computer and ensure the space vehicle's safe return to earth. This was the method that was available to the stranded astronauts aboard Apollo 13." (page 248)


Speaking of outer space, I give both the book and the movie, four stars.

Weather Forecasting

Well, guess what happened? Gonzaga lost. You might remember my post from a couple weeks ago. The Gonzaga's men basketball team was undefeated with four games to go. We figured out they had a 91% chance of remaining undefeated. They won the first three, then lost the very last game of the season. So much for the 91% chance.

It was a pretty good mathematics application, I thought. If this was presented to a math class, how would the students react. My guess is that they would say the math failed. A 91% chance is not a sure thing, but there have been some studies that suggest people take it to be that. And on the other hand, people assume that a really low percentage is the same as no chance. This might lead to a good class discussion.

This made me think about other forecasts? Specifically, how do weather forecasters decide on the percent chance of rain. I found a post by a weather person at WESH TV, Amy Sweezey. Although we haven't met, and I don't even know what she looks like, she seems delightful. And smart. I learned some interesting things about how they make a percent estimation of rain for the day.

So what does a 40% chance of rain mean? It turns out that it depends.

Let's say that there is a 40% chance of rain over about half of the area in question.

Some weather forecasts will call this a 40% chance of rain. In a way it is. There is a 40% chance of rain somewhere in the area.

Some weather forecasts will call this situation a 20% chance of rain. And again in a way it is. There is a 20% chance that it will rain where you are currently standing.

As Amy says, "When it comes down to it, you cannot base your plans around a rain percentage." It's more important to know where, what time, and how heavy it will be.

It's because of this that many stations won't even do a percentage. Instead they might use descriptive words like "scattered showers", "isolated", "a few showers", and "likely".

10 Cool Things about Srinivasa Ramanujan

I new very little about this person until recently. One way math teachers know about him is the fact that he often get a half page bio in math books. He is probably included so the company can show that their book is multicultural and perhaps that will lead to more sales. I hate to be cynical like that, but I noticed that once when on a textbook adoption committee. We had a sheet in which potential adoptions were given points in various categories. One was how multicultural it was. I guess it makes financial sense for the companies to load up multicultural references.

While that is a good idea, students often catch on to some of the overreaching. One story problem example was (yes, an actual example) "One day Running Bear was ..." Its fine, but pretty transparent what they are trying to do.

Back to Ramanujan (accent on the "nu" syllable and pronounced "new"). He didn't really do a lot of applied mathematics. Most of his work was in number theory and analysis.

He is featured quite a bit in the book The Music of the Primes. There is a book and a movie about his life The Man Who Knew Infinity. I recently saw the movie and I thought it was very good. And unlike many "biographical" movies, it didn't take liberties with the person's life.

Here are a few, I think, interesting tidbits of the life of Ramanujan:

1. Born and died in India, but much of his mathematical work was done at Trinity College in England.
2. Lived from 1887 to 1920, only 33 years.
3. In the movie he is played by Dev Patel, who was also the main actor in Slumdog Millionaire. G.H. Hardy was played by Jeremy Irons, who for me will always be the voice of Scar in The Lion King.
4. He had almost no formal training in mathematics.
5. He came to the attention of famous mathematician G.H. Hardy at Trinity College after writing to him.
6. Despite an invitation, he was at first hesitant to go as crossing the seas, according to his religion, could make him an outcast.
7. Suffered poor health all his life, but it was aggravated by the food at Trinity not matching well with his vegetarianism.
8. As his heath and depression became worse he attempted suicide by jumping in front of a London train.
9. A famous quotation is, "An equation for me has no meaning unless it expresses a thought of God."
10. A major coup was finding one of his lost notebooks in 1976.

Gonzaga Probabilities

A lot of games on TV will put up statistics way too fast. They're up for a few seconds and its tough to take it all in. Maybe it is the Detroit Lions' total rushing and passing yardage for each of the past three seasons. You got about five seconds to see it and try to make some sense of it. They obviously have some point they are trying to make with all those numbers, but tough to take in. Sometimes you just have to stare at things for awhile.

I did see one a few days ago that didn't have too many numbers in it. Gonzaga is currently the number one college basketball team. In fact they're undefeated. What are the chances they stay undefeated? It's 91%, they claim. They also showed the chance of winning each of their remaining games. Those chances are:

• 98%
• 99%
• 98%
• 96%

How did they come up with those numbers? I'm not sure. They didn't explain that. I tried looking on the internet to find out. That didn't work so well. Different associations (Vegas, ESPN, etc.) have their own different methods. Most of them let you in only partially on how they figure things. Turns out they take a lot of things into account. Your win-loss record, opponents win-loss record, playing home or away, point differentials, injuries, recent win-loss records, etc.

Then how they take all that data and come up with a percent is a mystery. What isn't a mystery is the 91% chance of winning all four of those games to remain undefeated. It is an "and" probability problem, which mean we can just multiply those individual probabilities together.

       (0.98)(0.99)(0.98)(0.96) = 0.9128 or just over 91%

On a related note, the University of Connecticut women's team won their hundredth game in a row last night. Pretty impressive. Impressive especially considering the following:

• A team that usually wins 80% of its games has a (0.8)¹⁰⁰ = 0.00000002% chance of winning 100 in a row.
• The best NBA record ever is Golden State last year. They won 89% of their games. Thus, a 0.00087% chance of winning 100.
• How about a team that wins 99% of its games? They have a 36.6% chance of winning a hundred in a row.

Here is a little higher mathematical problem. What kind of team would it take to have a 50% chance of going on a 100 game winning streak?

• x¹⁰⁰ = 0.5 
• 100(log(x)) = log(0.5)
• x = 0.9931

So you need to usually have a 99.31% chance of winning any single game you would play to have a 50-50 shot at winning 100 in a row.

In your mind this is maybe not the most important math application ever, but many students are into this kind of thing. And it does beat flipping coins and pulling various colored socks from drawers.

The Importance of Proofs

This blog is devoted to showing applications of high school mathematics. At first I thought the following topic wouldn't fit here, but then I decided it does. That is, the idea of proof. It really is tough for students to see the necessity of proofs. Usually encountered mainly in a geometry, they can be a real chore both for students and teachers.

Proofs are hard. So combine that with the fact that what is being proved seems pretty obvious, you get the double whammy of why students hate proofs - they are both hard and pointless.

Do we really need to prove that the sum of the first n odd positive integers is n squared. It wouldn't take to much time to get students to accept that fact without doing a proof. 

The square root of two is irrational. "I found the square root of two on my calculator. The decimals don't repeat. Yes. I'll buy it."

It seems that statements that are true for the first ten or so examples are probably going to end up being true. In fact, it's a little difficult to find situations where that doesn't hold true. If I make a statement that a coin I have will always come up heads, you might have your doubts. If I flip it ten time and I get heads each time you probably think that statement is, in fact, true, because it probably is a two headed coin.

I had heard quite a while ago that there was some statement having to do with prime numbers what was true way down the line, but then turned out not to be false at some point. I never knew exactly what that was, though.  

Well here it is. 

Suppose we want to find out if P is a prime number. Find 2^P and then divide it by P. If the remainder is 2, then P is a prime number. If it is something other than 2, P isn't prime.

I was a doubter at first. I did the first several and they all worked out, though. I'll take one example. Let's try the number 17.

2¹⁷ = 131,072
131,072 divided by 2 is 7,710 remainder 2
Thus 17 is a prime number.

This really works - all the way to 340. Then it doesn't.

2³⁴¹ is so big I can't tell you what it is. I could get my calculator to go as high as to the 331 power, then it refused to do any more.

Anyway, pretty amazing. It works for 1, 2, 3, 4, 5, ..., 340, correctly determining if the number is prime. Then this method claims 341 is prime. But it isn't, because 11 x 31 is 341.

There you go. Proofs aren't pointless after all. Still hard, but not pointless.

            *****************************************************************

Now, about last week's post. We worked out a score for the yet to be played Super Bowl of New England 30, Atlanta 27.

And how did that turn out? Pretty darn close I must say. New England 34, Atlanta 28.

The Possibility of Throwing a Football 100 Yards

Continuing from last week's post we are exploring the possibility of throwing a football one hundred yards.

That in a moment. But first, I heard on a sports radio show that for the upcoming Super Bowl game the Patriots are favored by 3 points. They also said that the over/under for total points that both teams would score was 57 (as I recall. It was a few days ago.) That is, you can bet that both teams combined will score more than 57 points or you can bet they'll score less. Of course I thought, "Hey, nice system of equations problem."

If x is the points scored by the Atlanta Falcons and y is the number of points scored by the New England Patriots, then we have:

y = x+3 and x+y = 57

That leads to x+(x+3) = 57
So, 2x+3 = 57
So, x = 27 and y = 30
Thus the odds makers are figuring:  New England 30, Atlanta 27.

Now, back to: Is it possible to throw a ball 100 yards?

You can look back on the last post to fill in what you might have missed, but currently we are at this formula.

d = (v²/g)ᆞ(sin(2θ))

The distance we are hoping to cover is 100 yards, which is 300 feet. The optimal throwing angle is 45 degrees. The acceleration due to gravity is 32. So,

300 = (v²/32)ᆞ(sin(2(45)))

Solving this gives us v = 97.98 feet per second which is 66.8 miles per hour.

I thought this should be possible. Ater all, there are people that throw a baseball over 100 mph. Surely someone could throw at 66.8 mph. Yes, footballs are heavier and probably have more air resistance, but still...

Turns out that 60 mph is about the best anyone is going to be able to do. You can check it out here
http://ftw.usatoday.com/2014/03/how-fast-football-throw-nfl-combine-logan-thomas. They've tested it out and no one has done better than that in the times they've been keeping track. Maybe someday someone will come a long that throws a football a lot faster than anyone else. But until then the longest passes will be well under 100 yards.

Throwing the Length of a Football Field

I had heard once that there was a quarterback (Roman Gabriel of the L.A. Rams) that could throw a football the length of the field. I think we need to investigate this. I found the formula for to calculate the range of any projectile:



We will work this out using the English System of measurement. If the football field was 100 meters long instead of 100 yards (It would only be 9 yards longer), the metric system would probably catch on in the U.S. a lot faster. Alas, that won't be happening any time soon.

So we let d = 300 feet, g = 32 feet per second, the angle is the optimum throwing angle of 45 degrees, the initial height is 6 feet, which I figure is about where the ball would leave the quarterback's hand. The velocity is what it would take to throw it that far.

This is not the way to go with this. For one thing, solving it is really hard. Trust me. I did it and its rough. You could assign it for a massive amount of extra credit, but otherwise it isn't worth it. Secondly, I didn't take into account that we are measuring to where it hits the ground which is not the same elevation that it took off anyway.

Let's just take a starting height of zero. That will make things easier and it turns out demonstrates an nice application of a trig identity.

So here we go. With a starting height of zero, the above equation becomes:

d = (vcos(θ)/g)ᆞ( vsin(θ) + vsin(θ))

d = (vcos(θ)/g)ᆞ(2vsin(θ))

d = (v²/g)ᆞ(2sin(θ)cos(θ))

But, there is a double angle identity that states: sin(2θ) = 2sin(θ)cos(θ). So our formula becomes:

d = (v²/g)ᆞ(sin(2θ))

This then becomes the formula that is commonly found for range of a projectile with a starting height of zero.

Nice application of an identity.

I feel that is quite enough excitement for one day. Next time we'll answer the question of whether Roman likely threw the ball 100 yards.

Dropping Pennies from the Empire State Building

You probably heard about the idea of dropping a penny from the Empire State Building and its

probability of doing some damage. The word from various sources is - No, it won't do much of any damage. If it hit you, you might think, "Hey what was that?"

They say (they being those in charge of such things) that a big part of the reason is the flat shape of the penny and the air resistance it would meet. Granted, but what if there is no resistance? Let's try it out.

(Final velocity)² = (Initial velocity)² + 2gd is a formula for a free falling object. (Time out - I, for the first time, used html to show exponents. Wow!! Big day for me.)

Mass of penny = 3.1 grams. (I'm going with that, but I saw that it is now 2.5 grams. And I didn't even notice.) Height of Empire State Building is 86.42 meters, and initial velocity of zero. That gives us a final velocity is 86.42 meters per second or 193.32 miles per hour. But as they say, that is a moot point as air resistance is not going to allow it to go that fast. Even if it is going fairly fast, it doesn't weigh very much.

Then I thought, what if you drop a baseball and what would it's impact be? And what would be a measure of its impact. Would we be looking at momentum or kinetic energy? If I had gotten an A instead of a gift B in physics, I might have an answer for that.

Momentum = mass x velocity
Kinetic Energy = mass x (velocity)²

So take the penny's momentum/energy. What would the velocity of the baseball be to have the same impact as the penny?

To make a long story short, I figured if having the same momentum, the baseball would be going 1.488 m/s or 3.329 miles per hour.

If looking at kinetic energy, the baseball's velocity would be 28.275 miles per hour.

Neither speed would hurt a whole bunch. Hopefully no one drops something like an anvil.

10 Cool Things about Rene Descartes

This is a blog about high school applications of mathematics. However, sometimes I've thrown in some historical things or bios. I was thinking maybe once a month I might focus on a particular mathematician. It might be interesting when presenting, say the topic of analytic geometry, some info on the so-called "Father of analytic geometry" Rene Descartes. We'll see how it goe.

I thought, also, instead of a dry bio, make it into a top ten list. These won't necessarily be the 10 most vital things about the mathematician. Just some things I found interesting.

Without further ado: Ten Cool Things about Rene Descartes

1. His hometown, LaHaye, France is now known as Descartes, France in his honor.
2. He earned a law degree from the University of Poitiers.
3. he did not publicize his theories on the solar system being heliocentric after seeing the treatment Galilleo received.
4. In spite of this, Pope Alexander still put his works on a list of prohibited books.
5. He never married, although had a daughter who died at the age of five.
6. Feeling there were to many things being accepted as true, he started with only "Cogito, ergo sum" - "I think, therefore I am.
7. At the turn of the millennium, the A&E Network conducted a survey of historians on the 100 most important people of the millennium. Descartes came in #32.
8. The concept of exponents predated his time, but Descartes had the idea of writing them as superscripts.
9. He showed that a viewers angle from the center to the edge of a rainbow is 42 degrees.
10. He wrote proofs verifying the existence of God.
11. 

World Population Growth

Here is an interesting graph. (https://ourworldindata.org/world-population-growth/) It shows the world's population growth up to the present day and then someone's estimates as to what will happen in the next few decades. It probably takes a little looking at for it to make sense. I combines two graphs in one. The horizontal axis shows the passage time in years and the vertical shows growth rates. The graph also shows the total population although these numbers are just recorded on the graph rather than being recorded on the vertical axis. As line graphs go, it's a pretty busy graph.

It seems to me that math teachers could make use of the graph in pretty much any high school mathematics class.

This actually started for me with information I found in the 2017 World Almanac. It showed population estimates going much farther back in time than this graph shows. You could look at that information as a set of ordered pairs with years being represented as x-values and world population (in billions) as y-values. The almanac states that in the year one the population was an estimated 300 million. That gave me an ordered pair of (1,0.3). Proceeding in this manner gave me ordered pairs of (1,0.3), (1250,0.4), (1500,0.5), (1804,1), (1927,2), (1960, 3), (1974,4), (1987,5), (1999,6), (2011,7).

Just using the raw data, an Algebra I class might simply write ordered pairs, or without seeing the above graph, choosing appropriately labeled axes to graph the data.

Higher math classes could look at finding an equation to model the data. I had more trouble than I thought I would. I guess that is because, as the graph shows, the rate has varied over time just in the last couple centuries, let alone millennia. Leaving out the first few ordered pairs and adjusting the data such as changing (1804,1) to (0,1) and so on, I was able to find an equation that had a correlation of r = .9647. Students could maybe experiment with similar things to get a best fitting curve.

Calculus students would be able to examine the blue population growth curve and discuss how it ties into first and second derivatives. It is interesting that the person making the future projections seems to think our current point in time seems to correspond to an inflection point. Students could discuss what that really means and mathematically and socially.