Mileage

I took a fairly long trip and I thought that made for a good and not extremely difficult application problem. It is a two-parter, with a little algebra, but not to hard.

I drove roughly 600 miles. I was on a freeway the entire time, so was getting a healthy 39 miles per gallon. The price of gas seemed was about $3.60 per gallon. How much did my trip cost me?

How much it cost me depends on how many gallons I used. So, how many gallons did I use? Since miles per gallon = miles/gallons, we have 39 = 600/x. It turns out that x = 15.39 gallons.

Again we can use a fairly obvious equation: cost per gallon = cost/gallons. Since the cost for each gallon is $3.60, we have 3.60 = x/15.39. This gives us x = $55.40.

Tycho Brahe - World's Coolest Scientist

I was reading the book Grapes of Math, written by Alex Bellos. In it he mentioned a couple of things about the scientist Tycho Brahe. Alex mentioned that he had a silver nose and that had a pet deer who fell to his death after drinking too much beer. What? I knew just a little bit about Mr. Brahe, but how did this information get past me?

Students often see mathematicians and scientists as being fairly plastic and dull. A few stories about Tycho could probably change their minds. He was not plastic. Part metal, but not plastic. Here are some top points about Tycho Brahe.

1. He was a nobleman and created something of a scandal by having children with a lowly peasant girl.

2. The silver nose thing seems to be quite true. Several sources back this up. As a twenty year old, he got into a mathematically induced fight with another student. It seems it was either over some mathematical point or over who was the better mathematician of the two. It did not end well for Tycho who lost a good part of his nose in a sword duel. It was stylishly covered with a metal nose piece.

3. There has been some debate whether the nose piece was gold, silver, brass, or copper. Tycho's body was exhumed in 2010. (For those into such things, the exhuming can be viewed on the internet.) One of history's mysteries solved - it was brass.

4. He did not agree with Copernicus. Copernicus died three years before Brahe was born. The Copernican theory was out there, but Tycho did not fully buy into it. Brahe was somewhere between Earth and Sun centered. He believed the Moon and Sun orbited the Earth and the other known planets orbited the Sun.

5. He built and made observations from the finest observatory in Europe. His observations were by eye as the telescope had not been invented yet.

6. He hired Johannes Kepler as his assistant. After Brahe's death Kepler went on to find the great important three laws of planetary motion, based on Brahe's observations.

7. The deer story seems to be true. He owned a tame elk as a pet. Tycho loaned him to a friend. One everning he got into some beer and became drunk. Later that evening, the elk fell to his death down a flight of stairs. (Feel free to insert any desired number of exclamation points.)

8. His death. There has been a theory that he was murdered by mercury poisoning, perhaps by a jealous Kepler. The exhumation in 2010 showed that he did not have elevated levels of Mercury. Johannes is off the hook.

9. It is probable that he drank too much at a banquet and felt it would be rude to dismiss himself. He died of a broken bladder.

Winning a car

I got my mail today. It turns out I might have won a car. First I have to scratch something off to see if I'm in the running. I scratched and it turns out I'm already a winner!!!! Looks like luck is on my side already!!! Now I know I've won at least something. There are only four choices. Three of them are great, and one is just a two dollar bill. So I'm thinking a three out of four chance of winning a big prize.

I checked out the fine print included in the add. It doesn't seem as good as it did a minute ago:

2015 Jeep Patriot or $25,000 - Odds of Winning - 1:25,000

$100 - Odds of Winning - 1:25,000

$2 - Odds of Winning - 24,998:25,000

So first of all, I didn't have a three out of four chance of winning something good as you had a chance to win the $25,000 or the car, but not both. So I have a two out of three chance of winning something good.

Secondly, it turns out I don't have a two out of three chance. I have a two out of 25,000 chance.

Good can come out of this, as there is much students can learn from this.

For one, it could be pointed out that despite what the ad says, these are probabilities and not odds, but why quibble?

Next, students could spot all the mind games that are being played. Getting you hooked by having you scratch off something. Feeling lucky when you are successful (which undoubtedly everyone is). Having what looks like three pretty good prizes, but is only really two.

Students can learn about expected value. The expected value could be found as $25,000 x 0.00004 + $100 x 0.00004 + $2 x 0.99992 = $3.00384

On the plus side, there is no way you can lose, but chances are you don't win very much. In most gambling, a person has to put up some money to play the game. That is the upside for the person/company/casino running the game. In those cases, the house typically sets up the game so the player's expected value is a negative number. What is the company's incentive in this case? They are obviously hoping to win their money on the back side of the game. They're hoping to get extra traffic to their showroom to sell a few more cars.

Gimmicky advertisements can be a great learning experience.

Deflategate math

There are a number laws relating to gasses that show a proportional relationship. One of these is Charles’ Law. Jacques Charles was the uncommon combination scientist/balloonist. He co-developed and rode in the first hydrogen balloon in 1783. Manned balloon flight was it its golden age and Charles played an important part.

Charles’ Law states that with pressure remaining constant, the volume of a gas are in direct variation. The experts say that tire pressure should be measured before driving as the act of driving will cause the air in the tire to heat up and thus expand.

You might be aware that a basketball left in the cold doesn’t bounce very well. That is because the decreased temperature means a reduced volume of air in the ball. If it is brought inside or run under hot water it will regain its bounciness.

A similar law, Gay-Lussac’s Law, states a directly proportional relationship between temperature (measure in degrees Kelvin) and pressure.

Some examples:

- A basketball is left outside at 8 degrees Celsius. Its volume was 444 cubic inches. Brought inside, it warmed up to 32 degrees. What is its volume now?

First of all, we need to be in Kelvins, so K = C+273, means the ball’s temperature went from 281 to 305 degrees Kelvin. Now setting up our proportion we see,

                                             444 : 281 = x : 305   →     x = 483 cubic inches

- NFL Rule 2.1 states that “The ball shall be made up of an inflated (12 1/2 to 13 1/2 pounds) urethane bladder…”. It is claimed that the Patriots football team has under-inflated the ball to try to gain an advantage. A league report claims that at least one of the balls was inflated to only 10.5 pounds per square inch. Suppose the Patriots then claim that according to Gay-Lussac’s Law the football would naturally lose pressure as it was brought from the warmth of the locker room to the field. If the locker room was 75° F (297.04° K) and the field temperature was 40° F (277.5944° K), might the Patriot’s claim that the football was actually within the guidelines?  

Our proportion comparing pressure and temperature could be written as:

                                            10.5 :  277.59 = x : 297.04 

We get an answer of 11.2 psi, which is still under the 12.5 minimum.

100,000 Times

I saw a commercial in which someone said, "My heart beats one hundred thousand times a day." I thought that sounded a little high. So in a week and a half it would beat a million times. That seemed high as well. We should check that out.

Sometimes students just aren't sure how to get started. One thing that can help to look at the labels. For example ,if someone wants to find out miles per hour, the label mi/hr would imply one would divide miles by hours. If someone had traveled at 23 miles per hour for 1.3 hours, how far would they go? The labels work out if the miles per hour quantity is multiplied by the number of hours. The labels would be (mi/hr) x (hr/1). The hours cancel leaving us with miles, just like we want.

Suppose a student wants to see if the 100,000 value is plausible, but has no idea how to begin. One could start with just looking at the labels and see if that works out. The Mayo Clinic says normal is somewhere between 60 and 100 beats per minutes. That is a pretty broad range. I have heard 72 in the past, so let's go with that. We want to see a final answer that is in beats per day. So the student is starting with beats per minute and we want to end up with beats per day. How do we get there? Hopefully they can see that 60 beats per minute will bridge that gap.

                    72 beats/min x 60 min/hr x 24 hours/day
                         = 72 beats/min x 60 min/hr x 24 hours/day
                         = 103,680 beats per day

Nicely done.

What number of beats per minute would correspond to 100,000 beats per day? As in the last problem, we can examine labels. This is not the only way, and not even necessarily the best way, but for some students it might make the most sense.

So we start with 100,000 beats per day and want to end up with so many beats per minute. We have to find some linking labels to get from one to the other. We have to fill in the blanks for:  100,000 beats/day x ?????? = x beats/min

                   100,000 beats/day x ?????? = B beats/min
                   100,000 beats/day x day/24hr x hr/60 min = B beats/min          
                   100,000 beats/day x day/24hr x hr/60 min = B beats/min
                            B = 69.444... beats per minute

Expected Value for Robbing a Bank

"When to Rob a Bank" is a book written by Steven Levitt and Stephen Dubner. It is a collection of stories from their blog. They are two economists who previously wrote the bestselling "Freakonomics".

An article in this latest book gives some statistics on bank robbery in the United States. It's not as lucrative a business as I thought. Bank robbers get away with it 65% of the time. Chances are, the average bank robber gets away with it, but there is a pretty solid chance he doesn't. Another drawback is that they don't get nearly as much money as I thought. The average haul is only $4,120. That is quite a bit of money I suppose, but it isn't going to make you rich. You would have to rob a bank a month to get yourself to a middle class income. And a lot more than that to get rich.

When I read this, I wondered what the expected value would be? Expected value is the average value you expect to gain in an experiment with a large number of trials. In this case, you have a 65% chance of making $4,120. But how do you put a value on getting caught? You would be going to jail, I'm guessing for roughly 5 to 10 years. How much would you pay to have your freedom instead? In other words - How much would you pay for a get out of jail free card? I'm guessing conservatively that has to be worth at least $10,000 to you. 

                    Expected Value = 0.65(4,120) + 0.35(-10,000) = -$822 

We've established mathematically that crime doesn't pay.

What if we didn't just rob one bank. Let's try robbing two. Basically three things could happen.

1. You successfully rob both banks. Probability = (0.65)(0.65) = 42.25%. Payoff = $8,240.

2. You rob one and then get caught trying to rob the second. Probability = (0.65)(0.35) = 22.75%. Payoff = -$10,000. We're assuming they won't let you keep the money from the first bank and you still are going to jail for 5 to 10.

3. Probability you are caught the first time. Probability = 0.35. Payoff = -$1,000.

                    Expected Value = 0.425(8,240)+0.2275(-10,000)+0.35(-10,000) = -$2,293.60

Crime doesn't pay.

Equal Temperatures

Students know that the same warmth registers differently on the Celsius and the Fahrenheit scales. They might be surprised to know that there is a point at which they are the same. That temperature could be found by using a system of equations. Celsius and Fahrenheit are related with the equation F=(9/5)C+32. Since we want to find out when the two scales are the same, we also need the equation F=C.

Initially, students might make up a table of values and notice that when it is warmer, temperature readings are farther apart -

                                 100 degrees Celsius = 212 degrees Fahrenheit
                                   40 degrees Celsius = 104 degrees Fahrenheit
                                     0 degrees Celsius = 32 degrees Fahrenheit

The differences are getting closer together - 112 degrees apart, then 64, then 32.

If we change the variables to x's and y's, the equations would perhaps look more familiar to students:

                                                 y=(9/5)x+32   and   y=1x+0

In this form students should see these are both linear equations, but have different slopes. While they might not know where it is yet, clearly those lines must have an intersection someplace. So, being straight lines, there must be one and only one temperature that is the same for each scale.

To get that temperature, the substitution method is probably the easiest way to go.

                                                               x=(9/5)x+32
                                                                5x=9x+160
                                                                  -4x=160
                                                                      x=-40

And so if x is -40, y must also be -40.

Being linear equations and already in slope-intercept form, algebra students should be able to easily graph them. Doing so, they would find an intersection point (or at least fairly close) of (-40, -40).