Water Towers

Do you know how water towers work? I have never known. I asked some people recently and they didn't know either. They're filled with rain water? It looks like a lot of water, but not nearly enough to last a city for very long. What happens if/when they run out? So many questions.

Here is a crash course that isn't going to cover everything, but it may answer a few questions you might have.

I thought the process kind of starts at the water tower, but it doesn't. The water we drink starts off at a water treatment plant. In a college biology class, we had a field trip to one. It was my only college field trip ever. Since I have all these questions now, I must not have paid very good attention.

Depending on the need, water is then pumped to the city or up into the tower. The water in the tower is just there in case it is needed. It holds a lot, about 50 swimming pools full, but still wouldn't last a city very long. It is probably enough for about a day. It can be used if power goes out, which would shut down the pumps. It can also be used for emergencies, such as fighting a fire. It is also used as a supplement if what is being pumped from the water treatment plant isn't enough.

As mentioned, it can be used in the case of a power outage. That is because water comes out from the tower by the force of gravity. That is why they are always located on the top of a hill. The water inside can only flow to points at a lower elevation than the tower itself.

There are plenty of good math applications involved here. Obviously, finding the capacity of the tower can be important. As can be seen, they might be conical, spherical, or cylindrical, or even combinations of shapes.

Finding the amount of pressure of the water flowing from the tower would be important. Every foot in height supplies 0.43 pounds per square inch. Thus, the higher it is, the more pressure. We'll look at these issues a little more next time.

Until now, I obviously never took the time to look into this topic. I got a lot of my information from several sources, but http://people.howstuffworks.com/water.htm was probably the best. That is where the diagram below is from.

A = From Treatment Plant
B = Pump
C = Water Tower
D = To Customers

The Bermuda Triangle

The Bermuda Triangle. It is the site of many shipwrecks and plane disappearances. Many dispute whether this is really all that extraordinary. After all, as can be seen, it encompasses a large area. Plus it is an area that has a lot of hurricanes and other disturbances in the weather. So all the mishaps could be something weird going on or maybe not that strange at all.

As mentioned, it is a large area. How large in fact. We have the standard A = 0.5bh formula, but it isn't easily used for this. So, this is a great application of Heron's formula.

To review, for any triangle with sides a, b, and c; what is called its semiperimeter is s = (a+b+c)/2.

The area then is A = square root of (s(s-a)(s-b)(s-c))

So, lets try it out. The value of s turns out to be 1,511. Plugging the numbers for a, b, c, and s into Heron's Formula gives a value of 437,600 square miles.

Different people give differing locations for the sides and vertices of the triangle, so there will be some differences. Also, we are not taking into account the curvature of the earth. Even so, it's a nice application.

Pronouncing Mathematician Names

Last time I took a shot at ranking the top mathematicians of all-time. One thing that has always bothered me is not knowing how to pronounce mathematicians names. Some are tricky. You might not know how to pronounce Euler or Euclid, but at least you figure they would be roughly the same. They aren't.

These names often don't get said aloud. If a person comes across a mathematician's name, its probably because of reading it. As a service, I decided to look up pronunciations of some of the trickier ones and list them here. If a math teach is going to talk about famous mathematicians, and I think we should, we at least owe it to them to get their names right.

I'm going with dictionary.com for these. If you think any of these incorrect, you'll need to take it up with them.

In no particular order, here we go:

Euler: OI-ler

Euclid: YOO-klid

Gauss:  gous

Descartes:  dey-KAHRT

Lagrange:  la-GRAHNZH

Leibniz:  LAHYB-nitz

Poincare:  pwan-ka-REY

Riemann:  RE-mahn

Pascal:  pa-SKAL or pa-SKAHL

Fermat:  fer-MAH

Galois:  gal-WAH

Bernoulli:  ber-NOO-lee

Cauchy:  koh-SHEE

Hermite:  her-MEET

Laplace:  la-PLAS

Fourier:  FOO-ree-ey

Huygens:  HAHY-guhnz

Top Mathematicians of All-Time

Who are the top mathematicians of all-time. Not exactly a math application, which is what I try to blog predominately, but I thought a quite interesting topic. I am no expert in this, but I consulted some websites of those that are. Or at least people that actually took the time to compile a list.

I had always heard the names of Newton, Archimedes, and Gauss to be the top three. They certainly deserve to rated highly, but not everyone puts them in the top three spots.

I found several websites. I thought I could compile these in some manner to come up with a fool-proof, master ranking. So, here we go.

This is from a website, http://www.eoht.info/page/Greatest+mathematician+ever that has their own ranking, then list several others as well. They (Their site is "Hmolpedia - an encyclopedia of human thermodynamics, human chemistry and human physics", no sure what that even means) have their own list.

Hmolpedia                   

1. Euler                
2. Gauss                
3. Newton             
4. Euclid
5. Archimedes
6. Descartes
7. Lagrange
8. Leibniz
9. Poincare
10. Pythagorus 

The-Top-Tens.com

1. Euler
2. Gauss
3. Archimedes
4. Aryabhatta
5. Euclid
6. Lagrange
7. Leibniz
8. Pythagorus
9. Descartes
10. Newton

Eells - 1962

1. Newton
2. Leibniz
3. Lagrange
4. Euler
5. Laplace
6. Euclid
7. Gauss
8. Archimedes
9. Descartes
10. Cardano

Bellos - 2010

1. Pythagoras
2. Hypatia
3. Cardano
4. Euler
5. Gauss
6. Cantor
7. Erdos
8. Conway
9. Perlman
10. Tao

Pickover - 2001

1. Newton
2. Gauss
3. Euclid
4. Euler
5. Hilbert
6. Poincare
7. Riemann
8. Galois
9. Descartes
10. Pascal

Allen - 1998

1. Newton
2. Archimedes
3. Gauss
4. Euler
5. Riemann
6. Euclid
7. Poincare
8. Lagrange
9. Hilbert
10. Leibniz 

The voting seems surprisingly close. Bellos, I'm sure a fine person, did seem to wander off from the rest of the herd a bit, but that's fine. So, let's use a 10 point scale, with first place getting 10 points, and so on. Here is the definite, not to be debated, beyond approach, all-time top ten mathematicians.

1. Leonhard Euler - 48
2. Carl Gauss - 45
3. Isaac Newton - 39
4. Euclid - 31
5. Archimedes - 26
6. Joseph Lagrange - 20
7. Gottfried Leibniz - 17
8. Rene Descartes - 11
9. Henri Poincare - 11
10. Bernhard Riemann - 10

Warriors 3-1 Comeback

I happened to be in the bay area this weekend. The bay area the home of the Golden State Warriors. A few days ago it wasn't looking good for them. Oklahoma City was ahead three games to one in their best of seven series. So to win, Golden State would have to win three games in a row.

There were a million sports radio shows that had their predictions of the likelihood of Golden State winning three in a row. Most all of them decided that it was not likely. But somehow they did. I like to think my presence in the bay area had a little to do with those wins. What were the chances anyway? Virtually all of the radio experts were making basketball predictions. However, it is also a good math application as well.

Theoretically I would think that 0.5 x 0.5 x 0.5 = 0.125 might be it. So the team with the lead has a 87.5% chance of winning the series. Historically, counting Golden State's win, the team that was ahead has now closed it out 95.7% of the time. It's close, but quite a bit ahead of our theoretical figure. But that does make some sense. We were assuming the chance of a single win is 50%. But if your team is ahead three games to one, you probably (but not necessarily) have the better team. Maybe that team against the same opponent would win 60% of the time. Then 0.4 x 0.4 x 0.4 = 0.064. That means the leading team would have a 93.6% chance of winning the entire series. Now we're getting there.

In fact, what winning percentage would correspond with the 95.7% figure? If the chance of winning three straight is .043 (1 - .957), then the chance of winning one game is the cube root of .043 or .350. That would mean that the team ahead in the series would normally beat the other team in a single game 65% of the time. That seems fairly reasonable.

I found a bunch of cool historical stats like this on the following website: http://www.whowins.com/tables/up31.html

I'm sure there are others like this with different scenarios. This one states the chances of winning if a team is up 3-1 in the playoffs in various settings.

• In the NBA it is now 95.7%. 
• In major league baseball it is only 85.2% (slightly less than the theoretical percent).
• In the NHL it is 90.1%
• In all three sports combined it is 91.7%

This might be of a bit more interest to students before the series is over, but regardless, I think its a pretty cool application.

More Batting Average

Continuing from last week's math application, we know that a player's batting average is found by dividing his hits by his times up to bat.

There was just a news article how Jon Lester, a fine pitcher, but poor hitter for the Cubs has started the season at 0 for 59. In batting average talk this is not called "zero", but .000. This is a record to start things off this badly.

Of those getting up to the plate over three hundred times, the record is Dean Chance at .066. Another poor batter was Fred Gladding who went one for 63.

Students might have heard that Ted Williams was the last to "hit four hundred". This means during that particular season (1941) his batting average was over .400. He was 185 for 456 which is .406.

Students will have heard of "batting a thousand". This means a hit every time at bat, which is mathematically actually batting "one", but it is written as 1.000 and in baseball it is called batting one thousand.

Here is batting average problem with a system of equations. You start rooting for a player that mid season has a batting average of .312. From that point you keep track and he goes 40 for 100 and ends the season with a batting average of .341. How many hits and at-bats did he have when he was hitting .312?

x/y = .312   and    (x+40)/(y+100) = .341.

Even rounding off to thousandths place there will be some round off error and final answers may need to be adjusted a bit. Obviously, at-bats and hits will have to be whole numbers.

Batting Averages

Batting averages are a nice math application for those starting out learning algebra. Some students will think this is a pointless application, while others will think, "Hey, this stuff is useful after all."

A player's batting average is found by dividing the number of hits by the times at bat. It is rounded to the nearest thousandth place. It is never pronounced as it should be. While 0.250 should be pronounced "two hundred fifty thousandths", baseball people pronounce it "two-fifty". A person with a batting average of 0.267 is said to be hitting "two sixty seven". And so on.

If two of the pieces are known, the third can be found with a little algebra. As of this moment, these numbers are current numbers for these players:

David Ortiz has 41 hits in 128 at-bats. What is his batting average? (41/128 = x; x = .320)

Zach Cozart has 113 at-bats and is hitting .319. How many hits does he have? (x/113 = .319; x = 36)

Eric Hosmer is batting .336 with 47 hits. How many times at bats does he have? (47/x = .336; x = 140)

Also of interest might be the fact that sometimes thousandths place isn't enough accuracy. In 1949, George Kell won the batting title over Ted Williams even though they were both listed as batting .343. George was 179 for 522 (0.3429) and Ted was 194 for 566 (.3428).

Pretty much the same thing happened in 1970, with Alex Johnson (.3289) defeating Carl Yastrzemski (.3286) by a whisker.