Leaves

 I looked out upon my lawn covered with leaves and wondered how many there were. The brute force method is to go count them. Lets say I don't have the time or inclination to do that. I could take some smaller areas of the lawn, count the leaves there, and compare them to the whole.

This is the same kind of thing that is done in surveys. DVD sales, TV ratings, unemployment numbers, all survey to get their results. We recently had a presidential election. It took a while, a long while, because they said they were getting to a 99.5%. That doesn't give you certainty of a number. It just means you are 99.5% certain of being within a range of numbers. Almost all nationwide statistics are surveys. When we don't do that - the census every 10 years - it takes months and months to get to everyone.

I'm sure you know how this kind of thing works, but I took the time to do it, so I feel obligated to share.

I (roughly) marked one yard by one yard squares. There were three of them and they averaged having 125 leaves in each. Then I found the area of the entire yard. Nicely, it is pretty much a trapezoid. The bases about 7 and 14 yards with a height of 7 yards. That gives an area of 73.5 square yards.

Next, a proportion would be 125/1 = x/73.5. So, x = 9187.5 leaves. 

So, a nice problem for maybe an Algebra I or general math class. It can lead into the idea of the surveys that go on every day.

The Lighthouse at Alexandria

I was watching a video about the Lighthouse at Alexandria. It no longer exists, but was one of the seven wonders of the world. The video said it could be seen from thirty miles away. They didn't mention how tall it was. That can be figured out, though. There is a formula. To the right is a diagram of the situation.

Suppose v is the height of the lighthouse. OH is the distance to the horizon from the top of the lighthouse. Using a geometry thereom, the diameter squared = (diameter+h)h. You could also come up with a formula by using the Pythagorean Theorem. In that case, (v+r)squared = r squared + OH squared. And then the radius or diameter of the earth could be substituted in. Another extension could be to find formulas for distance to the horizon for other other planets. Formulas that are then derived usually are simplified by the fact that the value of v is so small compared to r. 

Those formulas can be found on-line. Using one of them, by my calculation, to have the horizon be 30 miles away you would have to be 605 feet up in the air. Pretty tall. The Eiffel Tower is 984 feet. But the tallest now are less than 300 feet. So the video, or I seem to be off. But maybe not. It's certainly possible. You don't get picked as a wonder of the world if you're not pretty impressive.

Going for Two

Seems like I wrote something like this before, but things have changed now, so it is time to revisit. In the NFL, after you score a touchdown, you can go for a one-point conversion, by kicking, or go for a two-point conversion, by running or passing. The two pointers are harder to make, though. So which should you do? The kicks pretty much always went through. Because of that, the NFL decided they needed to make things a little more interesting and moved the kick back a ways. It still almost always goes through, but the percent is down a little bit - to a success rate of 93.8% of the time. The pass or throw option is roughly half - a success rate of 50.1%.

We should find the expected value for each. 

For a kick, you can get one point, 93.8% of the time:

                    1 x .938 = .938 points per try

For running or passing, you get two points, 47.9% of the time:

                    2 x .501 = 1.02 points per try

Is that even worth messing with? It depends on how good your team is, but typically a team scores something around 60 touchdowns in a season. That would mean roughly (.938 x 60 =) 56.28 points if you kick all the time. And (1.02 x 60 =) 61.2 when running or passing. So about five point over the course of a season. So not a lot, but then again, it only takes one point to lose a game.

And of course your decision could depend on the situation in any particular game. If you score a touchdown near the end of the game and you are now behind by two, you definitely would go for the two-point attempt. 

By the way, I don't know if this was in the book, but Scorecasting is a great little book about sports and how coaches don't always do what makes the most sense. 

Moving Averages

 

Moving averages are in the news thanks to the Corona virus. They takes the major fluctuations out of data that might otherwise jump around a lot. The following graph shows this. There are fluctuations during the week because of the nature of the work week. I'm not exactly sure why, but maybe Mondays usually show more positive cases than Fridays. The following graph shows this situation in Georgia, but I think this kind of thing can be found in pretty much any state. And after that is some information from Investopedia. 

Understanding Moving Average (MA)

Moving average is a simple, technical analysis tool. Moving averages are usually calculated to identify the trend direction of a stock or to determine its support and resistance levels. It is a trend-following, or lagging, indicator because it is based on past prices.

The longer the time period for the moving average, the greater the lag. The 50-day and 200-day moving average figures for stocks are widely followed by investors and traders and are considered to be important trading signals. 

Moving averages are a totally customizable indicator, which means that an investor can freely choose whatever time frame they want when calculating an average. The most common time periods used in MA's are 15, 20, 30, 50, 100, and 200 days. The longer the time span, the less sensitive the average will be.

There is no correct time frame to use when setting up your MA's. The best way to figure out which one works best for you is to experiment with a number of different time periods until you find one that fits your strategy.

Moving averages, a.k.a. running or rolling averages, that I've see, usually use 3 or 7 data points at at time. I had no idea as many as 200 were ever used.

I randomly picked a baseball player from the past and found his number of home runs each season. Incidentally, I once heard a baseball stat man say it would take about three seasons to get an accurate picture of how good a batter is. Now that I think about it, that makes sense, At about 500 at-bats a season, that would be 1,500 at-bats total. I saw on-line that the Pew Research center will typically survey 1,500 people at a time.

Anyway - here are Mickey Mantle's home run totals each season. Then a rolling average, three years at a time, then five years at a time. Notice, it gets a little less bumpy each time.

13, 23, 21, 27, 37, 52, 34, 42, 31, 40, 54, 30, 15, 35, 19, 23, 22, 18

            19, 24, 28, 39, 41, 43, 36, 38, 42, 41, 33, 27, 23, 26, 21, 21

                        24, 34, 34, 38, 39, 40, 40, 39, 34, 35, 31, 24, 23, 23

Fires

There has been a lot of fire action in the West the last couple of weeks. And as time went by there was a lot of smoke. I got an air quality app. I was wondering why the graph chopped off at 500. My app said we were well into the 500's. I learned that that was as high as the chart goes. We were literally off the chart. I guess they assume you aren't going to have numbers like that. Our house was never in imminent danger of fire. They had a map on the news that had green areas meaning "Get prepared to maybe leave", and yellow was "Get ready to go, and red was "Go now". Our house looked to be 5 to 10 miles from the green and maybe 10 to 15 from the yellow. So, a bit concerning, but not too bad. I figured we're in a city, not a forest. I guess I never really paid attention to how many trees there are on our street. There are a disturbing number.

But everything is much better now. When I was in high school / college I had a summer crap job of working in a youth parks program. That is when I learned I was not cut out for manual labor. We built trails and dug ditches and paved a parking lot. Yes, I was literally part of paving paradise and putting up a parking lot. Its not as much fun as it looks like. My job in paving was to be above where the tar comes from the dump truck, down a chute. I was standing above that bin and was to keep the tar moving along with some kind of stick they gave me. It was about 105 degrees those days and the tar itself was about a billion degrees. And if you've smelled fresh tar before you know its not pleasant. I spent hours right on top of that heat and smell.

Anyway, for about a week there was a big forest fire that they used us for to help "mop up". That was kind of fun. One time we were kind of on the front lines, but otherwise we just had a hoe or shovel and found smoldering places and put those out. We also had back pack filled with water and a squirt gun thing attached. That was super fun.

I'll never forget the lunches. I hope they've improved. Some kind of meat sandwich with almost no mayonnaise and an 8 oz can of unsweetened grapefruit juice. That was nasty. I was maybe the thirstiest I'd ever been. But, I lived and had an adventure, so that was cool. 

Tipping

 We'll I've gotten back on the horse and ready to do more math application blogs. I took a few months off. Not for any particularly good reason, although I did move. Piece of advice - Try not to move. Or at least make enough money to afford to pay one of those companies that will do everything for you.

Speaking of that, how much should you tip them? I don't know if you actually tip moving people, but its a transition (no idea how to spell "segway") into my post. 

First of all, I have a thing about tipping. The really important people don't get tipped - doctors, emt's, teachers, firemen. Its mostly people in a restaurant that take your order and your money. As restaurant workers go, they really aren't the hardest workers there. The cooks aren't getting tipped. The bus boy isn't getting tipped. The night time custodian isn't getting tipped. Plus, I'm even more annoyed when the credit card machine asks you how much you want to tip. I had that happen at a place where I got my own drink and all the server did was hand me a donut (that someone else made). Does that really deserve a tip?

Anyway, they say 10% isn't enough anymore. So now I'm not just bothered, I'm mathematically bothered. Apparently, tipping needs to go up to at least 15% because of inflation. That is either greed or a lack of understanding of how percentages work. 

Something used to cost $100 dollars, and lets say now it costs $140. A 10% tip would have meant a $10 tip. Present day, your 10% tip is $14. Yes, there is inflation, but the percent takes care of that. If prices go up and you still tip at 10%, the amount of your tip goes up. And what would 15% of $140 be? $21.

At cash registers that give you choices of tips, 15% seems to be the low end. It should still be 10% if it even exists at all. 

Whose idea was this? I'm guessing the people getting the tips. I don't want to be rude, but I you did betting in school you probably would understand percentages better and you wouldn't be working as a server in a restaurant to begin with.

Note - Future blogs will not be as mean spirited.

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.