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).

"Large" Soft Drinks

I was in a local fast food restaurant, who will be left nameless. Anyway, after getting my Mcdrink I thought it seemed a little small. It was called a "large" after all. So I was thinking it should be at least a 32 ounce or 40 ounce drink. Being something of a connoisseur of these things, I felt this was well off.

It reminded me of a time when I verified to a class the measurement of a supposed 32 ounce cup. It was like an episode of Mythbusters. It turned out to indeed be the as advertised 32 ounce container.

I began my investigation by Googling "volume of cup sizes" which led me to websites that were not even close to helpful. Another approach was necessary. The plastic cups found in convenience stores or fast food restaurants are close to, but not quite, cylinders. They slope somewhat so that the top of the cup is a little wider than the bottom. Mathematically, this could be called a truncated cone or a frustrum. To find the volume we multiply the area of the base by the height. The area of the base can be found by using the average of the radii of the two bases.

Getting as close as I could on the measurements, I found radii of 4.85 cm and 3.15 cm and a height of 16.3 cm. We then have an average radius of 4.0 and a total volume of (Pi)(4)(4)(16.3) = 819.3 cubic centimeters. One cubic centimeters is equal to 0.033814 fluid ounces, so we can multiply to find a volume of 27.7 ounces of fluid.

While I know the term "large" is relative, I was a little mcdisappointed nonetheless.

John Nash

As my first post, let's look at the life of the late John Nash. He and his wife Alicia were tragically killed in a traffic accident in May of 2015 - just about a week ago from when I'm writing this. For students, he might be the mathematician they are most aware of, being the focus of the best-selling book and Academy Award winning movie, A Beautiful Mind.

He was born in Bluefield, West Virginia in 1928. He showed early promise, even skipping a year in school. He graduated from the Carnegie Institute of Technology and then obtained his Doctorate from Princeton at the age of 22. He then worked of the Rand Corporation and taught at M.I.T.

He majored in mathematics, specifically focusing on game theory. Is it in a country's best interest to go to war? Should my team pass or run the ball? What should my investment strategy be? Should we drop our prices and attempt to undercut our competition? Should I buy more land or put motels on the land I have? Game theory, and especially the concept of the Nash Equilibrium, have applications in many different areas. It is for game theory that Nash would later win his Nobel Prize.

Interestingly, his and Albert Einstein's paths crossed at Princeton. Nash presented some of his thoughts on relativity to Einstein. The meeting wasn't totally satisfactory. Einstein let Nash know he needed to go learn some more physics.

His whole life, John Nash had always been what kindly could be called eccentric. His behavior became increasingly bizarre resulting in a diagnosis of schizophrenia. The National Institute of mental Heath's website states that, "People with the disorder may hear voices other people don't hear. They may believe other people are reading their minds, controlling their thoughts, or plotting to harm them. This can terrify people with the illness and make them withdrawn or extremely agitated."

Image courtesy Wikipedia.org

John Nash certainly seemed to fit that description. Thankfully, in the 1980's, he began showing improvement. It turns out this improvement was not as miraculous as some have assumed. The literature shows that ten years after its onset, approximately 25% of schizophrenics are "much improved". After thirty years, this increases to 35%. The chances of recovery seem improved with having a home, job, and hope. Thanks to Alicia and the support of colleagues John had these.

In his later years Nash was granted a number of honors including his Nobel Prize in 1994. He continued to work a Princeton University for the rest of his life.