Measuring Marajuana

I heard on the radio that someone was arrested that had 127 pounds of marijuana in his car. I wondered if that was even possible. Of the little I know about it, it is really light, so 127 pounds worth would probably take up a lot of space.

So what might that look like? Would it fill a shoe box? A trunk? The entire car?

Note: If you do this exploration with your math class, you probably need to be careful in how you frame this. If you present this as doing this as research as a member of law enforcement, you should be fine.

Anyway, I started my quest by going on-line. I found some websites that seemed to be in the know. The first person I read about had pretty much my same question regarding the relationship between volume and weight of marijuana.

His was a short post - about five sentences. I counted 48 typos. Not good, - e.g. "i duno wher im goin. somone help me out tho is there a way i cud mesure cannabis with a mesuring cup". I say about 5 sentences because punctuation did not seem to be his strong suit. On the plus side, he does know how to correctly spell cannabis.

A lot of people said this all depends on a number of different factors. I'm sure it does, but all I'm looking for is an estimation. One said an ounce is about a ziplock bag full. Another said he got an ounce that was about 3"x 3"x 2". They seemed like they were in the same ballpark, so lets go with that.

• So, a one ounce weight is 3"x 3"x 2" = 18 cubic inches.
• There are 16 ounces in a pound, so 1 pound is (16x18 =) 288 cubic inches
• 127 pounds must be (127x288 =) 36,576 cubic inches
• This would be easier for me to picture in cubic feet. There are 12x12x12 = 1728 cubic inches in a cubic foot, so  36,576 / 1728 = 21.17 cubic feet.
• The cube root of our answer is 2.8

So a box that is 2.8 feet on each side would hold this guy's marijuana. So you might be able to get that in your trunk. It certainly fits in the back seat of your car, with plenty of room left over for your other crime paraphernalia. But I don't know if you want that in your back seat. Maybe it didn't fit in his trunk, so he put it in plain sight in his back seat and that is what caused him to make the news.

Running Pace

I ran in a 3.1 mile race yesterday. My time was 32 minutes 10 seconds. You're right. That isn't very fast. I used to be fast. In fact, my goal now isn't to have my all-time best, but to beat my all-time best doubled time. If I ran a four minute mile at some point in my life, my goal now would be run a mile in 8 minutes. That keeps my interest up and helps me to believe I'm still competitive.

A question any runner might ask is, "What was my pace?" By pace a runner would want to know how many minutes per mile was he or she running.

This type of problem would be easier if there were a hundred seconds in a minute. In this case 32.10 divided by 3.1 would give me my pace. Alas, it isn't like that. I believe it was the Greeks that messed things up for us, favoring the number 60 rather than 10 or 100 or some convenient number like that. So if I'm going to divide, I have to take a little more care in doing so. I didn't run 32.10 minutes. I ran 32 10/60 minutes.

Here are my steps:

• 10/60 = .1666...
• My time for the race was therefore 32.1666...
• My division is 32.1666 / 3.1 = 10.376 minutes per mile

Technically correct, but most runners would want it in minutes and seconds per mile. The 10 minutes part is fine. The 0.376 minutes can be converted to 0.376 minutes x 60 seconds/minute = 22.6 seconds.

My blazing pace was 10 minutes 23 seconds per mile.

This is a good application in itself. It is also a trigonometry application. Those pesky Greeks used the number 60 for dividing up angles as well as dividing up time. As in, 30.73 degrees is how many degrees and minutes, or even, how many degrees, minutes, and seconds? Or perhaps you have degrees, minutes and seconds and you just want the angle measure strictly in degrees.

By the same method as the above problem, 30.73 degrees = 30 degrees, 43.8 minutes = 30 degrees, 43 minutes, 48 seconds.

If the Greeks had decided that each degree is made up of 100 minutes, it is a much easier problem - 30.73 degrees = 30 degrees, 73 minutes.

Intentional Fouls

There is a basketball statistic known as offensive efficiency (OE). It is the number of points a team score in one hundred possessions. The top teams as of right now is Golden State with and OE of 113.2. Worst is the Philadelphia 76ers at 94.5. They happen to also have the best and worst records, respectively, in the league.

In the basketball world there has been a debate about intentionally fouling a team's worst foul shooter, let him shoot two foul shots with the idea he is probably going to miss one or both of them. Is this a good strategy? Some math can be used to try to find an answer to this.

This debates centers mostly around Andre Drummond of the Detroit Pistons. he's good at some basketball things, but shooting free throws isn't one of them. He currently makes 34.9% of them.

A good probability problem might be, What is the probability makes at least one, i.e., he makes the first (event A) or second (event B) shot?

Pr(A or B) = Pr(A) + Pr(B) - Pr(A and B) = 0.349 +0.349 - (0.349)(0.349) = 0.576 = 57.6%

But I digress. Let's answer the specific question of how would Andre do just shooting free throws each time down the court? The probability of him:

Missing both = 0.651 x 0.651 = 0.424
Miss the first and make the second = 0.651 x 0.349 = 0.227
Make the first and miss the second = 0.349 x 0.651 = 0.227
Making both = 0.349 x 0.349 = 0.122

His expected value of scoring for a possession is:
0(0.424) + 1(0.227) + 1(0.227) + 2(0.122) = 0.698

Per one hundred possessions that would be 69.8 points. That compares to the Pistons team OE of 102.5. So yes, based on this, foul him. However, there are other factors to consider. Each player can only commit six fouls before being disqualified. You may have to pick your times. That is what the opposing teams have been doing.

How about the second worst free throw shooter in the league - DeAndre Jordan at 42.1%? Doing the same math, in a hundred possessions, his free throws would account for 84.2 points. His team, the Clippers, usually score 106.2, so again - foul him.

The third worst shooter is Dwight Howard. his free throw shooting would gain 109.8 points. His team's (the Houston Rockets) OE is only 104.2.

By my calculations, this fouling strategy would only be effective with two players in the entire league.

Aspiring NBA basketball players - practice your free throws.

Revolutionary War Cryptography

Spies have been around for a long time. Part of being a good spy is being able to send and receive coded messages. Lately, mathematicians have had a major role in trying to break these codes. There was a major movie, The Imitation Game, about mathematician Alan Turing and his breaking of the German Enigma code in World War II.

Codes go way before that, of course. I'm reading George Washington's Secret Six: The Spy Ring that Saved the American Revolution. The title seems a little overstated, but then again, I'm not done with the book yet. There are a couple of interesting items.

Invisible Ink - I always thought that was a made up thing. The author of the book says,

"The practice of writing with disappearing inks was nothing new. For centuries people had been communicating surreptitiously through natural and chemically manipulated inks that became visible when exposed to heat, light, or acid. A message written in onion juice, for example, dried on paper without a trace, but became readable when held to a candle. Secret correspondence in the British military often had a subtle F or A in the corner indicating to the recipient whether the paper should be exposed to fire or acid to reveal it message."

Interesting. Also, the coding itself was not as complicated as it is now, but still pretty effective. The book says that Benedict Arnold, when communicating with his British contact used,

"invisible ink and a book-based code. He based his code on two books: William Blackstone's Commentaries on the Laws of England and Nathan Bailey's An Universal Etymological English Dictionary. Each word was denoted by three numbers separated by a period. The first was the page number, the second was the line, and the third was the position of the word, starting from the left margin, in that line. for example, 172.8.7s stood for "troops": page 172, line 8, seventh word in. The s at the end simply made it plural."

The method is very clever. Although, if he was clever enough, Benedict wouldn't be so well known to us today.