Math and biology: tree growth

Recently in Victoria, the tutor noticed a sequoia tree in front of the Parliament Buildings.  The plaque said it grows, on average, 1 foot per year. Is that exponential growth?

I’ve introduced exponential growth in a couple of contexts: see my articles here and here. Its basic attribute is that with exponential growth, the quantity increases by a fixed percentage every year, rather than a fixed amount.

Growth of a foot per year is not exponential, by that description. However, natural growth processes almost always are. How can we reconcile the situation?

The answer is that growth is increase in mass, not height. Mass, in the case of the tree, is directly connected with its volume. So the question is, if its height increases by one foot per year, is its volume growing by a fixed percentage per year?

For today’s post, we’ll define the (mathematical) parameters of the situation and get some initial numbers.

Let’s make the simplifying assumption that the tree is a cone. I saw it in the dark, but I’d say it might be 10 feet across (radius=5). The plaque says it’s 100 feet high. The formula for volume of a cone is

V=πr^2h/3

where r is the radius, while h is the height.

Right now, the tree’s volume is approximately π(5^2)100/3=2618ft^3.

Since its diameter is about 1/10 its height, we’ll imagine the tree’s diameter grows 1/10 of a foot per year, so its proportion of diameter to height persists.

Next year, the tree should be 101 ft tall and 10.1 ft across (radius 5.05 ft). Its new volume will be

V=π5.05^2(101)/3=2697ft^3

In absolute terms, the tree’s growth, this year to next, will be (2697-2618)=79ft^3. Its percentage growth will be 79/2618=3%.

If its growth is 3 percent every year, the tree is growing exponentially. Next post, we’ll investigate whether we can confirm it:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: arithmetic mean

Tutoring math, you see topics come in and go out of style.  Arithmetic mean is such a concept….

 
In math, the “mean” is known by some as the average. To find the mean of 65 and 83, add them and divide by two:

mean=(65+83)/2=148/2=74

Between 65 and 83, 74 is an arithmetic mean: it’s in the middle between 65 and 83.

What if you were asked for two arithmetic means between 65 and 83? They would be two numbers that offer a path from 65 to 83 in three equal steps, as shown:

65,mean_1,mean_2,83

The approach is to imagine the distance covered from number to number in the sequence above. Let’s call the distance d:

65,mean_1,mean_2,83=65,65+d,65+2d,65+3d

Therefore

65+3d=83

Subtract 65 from both sides:

3d=18

Divide both sides by 3:

d=6

Therefore, the distance of each “jump” in the sequence

65,mean_1,mean_2,83

is 6. The sequence is

65,71,77,83

The two arithmetic means between 65 and 83 are 71 and 77.

Historically, arithmetic mean is a term often seen on exams; it comes and goes. Now, if you do see it, you’ll more likely be prepared:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Ethanol: viable as a transportation fuel?

Continuing about autos, the tutor turns the ignition on another topic: ethanol as a fuel in North America.

Sometime in the 2000s, if not before, ethanol’s use as an alternative auto fuel became a hot topic. Its use in Brazil is famous; why not in the US?

As I understand it, a couple of motivations spurred the use of ethanol:

  1. Possible ecological benefits.
  2. Relieving the high US expenditure on imported fossil fuels.

The first motivation, I believe, has come under serious question. For instance: some people mistook “renewable” to mean “reduced carbon footprint”; the two concepts aren’t necessarily related. Wood, for example, is a renewable fuel; yet, burning wood to meet America’s energy needs would probably not be ecologically sound.

The second motivation is possibly more rational. However, to make a fair judgment, one must realize that ethanol has only two thirds the energy of gasoline; therefore, the price of ethanol needs to be less than two thirds the price of gasoline for ethanol to be viable.

Today, the price of gasoline is $3.43/gallon, according to fuelgaugereport.com. The price of ethanol, at $2.17, (dtnprogressivefarmer.com) is less than two thirds the price of gasoline. Therefore, today, ethanol makes economic sense as an alternative to gasoline – theoretically, anyway. However, the prices of both fuels are volatile.

In future posts I’ll be exploring the ecological dimension of ethanol as a gasoline alternative as well as its economic aspect.

My children await their return to school; all the best to those trying to reach a settlement:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

English: compound words

From essay writing, one learns to define one’s own terms.  During the weekend before regular tutoring recommences, the tutor discusses a topic his own children have been exploring.

My (smart aleck) son recently told me, “Dad, important is a compound word.”

“No, it’s not,” I replied.

“But it is though:  import…ant.”

I told him that, although import and ant are both words unto themselves, important is not a compound word.  My reasoning:  a compound word’s meaning combines the meanings of the words it contains.  That’s why something is a compound word: it means some thing.  Overpass is another example.

Important is not a compound word, I continued, because import ant is meaningless – unless, perhaps, you’re in the business of ant importation.  Even then, its meaning isn’t the same as important.  A true compound word has the same meaning even if the words are said separately.  That’s my definition.

I sought back-up for my definition in three dictionaries.  Those that commented at all, tended to side more with my son:  that a compound word simply is made from two standalone words.

This fun little post has opened up a few great topics. I’ll be happy to continue them in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Humidex: a Canadian invention

The tutor enjoys discussing scientific concepts, particularly ones with a Canadian dimension.

The humidex provides a meaningful measure of a day’s heat. By considering dew point and temperature together, it provides an “effective temperature” that suggests the comfort level of the conditions. The “effective temperature” is referred to as “the humidex”.

The humidex is calculated as follows:

H=T+0.5555(6.11e^(5418*(1/273-1/dewpoint))-10)

where T is in Celsius, while dewpoint is, importantly, in Kelvin. To get Kelvin, you add 273 to the Celsius temperature. Therefore, a dew point of 20C is 293 K.

I have verified the formula using the following data set:

temp (°C) dewpoint (°C) humidex source
30 25 42 Wikipedia
28 20 36 Planetcalc: Humidex

When the dewpoint is 10C (283K) or less, the humidex is virtually the same as the Celsius temperature. For day-to-day life in Canada, it’s only on hot summer days, when the dew point can rise into the high teens or above, that the humidex becomes important.

Today’s humidex formula comes from J.M. Masterton and F.A. Richardson, Canada’s Atmospheric Environment Service, 1979. However, the humidex was first used in 1965, possibly with a different formula.

Source: csgnetwork.com

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Social Studies: How Upper Canada became Ontario

The tutor spends most of his time on math and science.  This particular question has never been asked during tutoring; rather, it comes from personal curiosity.

While I’ve long known that Upper Canada is now Ontario, while Lower Canada is now Quebec, I’ve always wondered how the names Ontario and Quebec came about. The names Upper Canada and Lower Canada are easy to understand, since they relate to progress of the St. Lawrence River:  Upper Canada was proximal to its upper course, while Lower Canada straddled its lower course.

Due to problems chiefly in Upper Canada, it was merged with Lower Canada in 1841, resulting in the United Province of Canada.  Ontario and Quebec emerged thence in 1867.

Ontario, according to Wikipedia, is named after Lake Ontario; the name originates in either the Huron or Iroquois language.  Wikipedia also informs me that Quebec is the Algonquin name referring to the environs of Quebec City.

Clearly, much more needs to be discussed (in this blog) about the history of Canada. I look forward to livening it up in future posts:)

Sources:

Bowers, Vivian and Stan Garrod. Our Land: Building the West.  Toronto: Gage    Education Publishing Company, 1987.

Wikipedia: Province of Canada.

Stanford, Quentin H.,ed. The Canadian Oxford School Atlas, 6th edition. Toronto:           Oxford University Press.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Random numbers, continued

Tutoring math, computer science might always be around the corner.  The tutor continues the discussion of random numbers, including a “live” example.

As I mentioned in my previous post, random numbers comprise a favourite topic among computer science people. At its core is the paradox of a random number being produced by a calculation device.

The definition of random, from the point of view of statistics, is that every possible value has an equal chance of being selected. For example: imagine you write each number from 1 through 100 on its own (equal-sized) slip of paper, put the slips all into a hat, then have someone reach in (without looking) and grab one of the slips of paper. We’d likely agree that’s a random selection. The person drawing the number, we believe, would be equally likely to pull out any of the slips of paper, hence any number between 1 and 100.

The experiment above is trustworthy because we can picture it and empathize with the person drawing the number. We know that, not looking at the hat while pulling the number from it, we ourselves would be hard-pressed to control the number drawn. When a computer produces a random number, however, we may not know how. The question inevitably arises, “Can we be certain the number is random?”

To offer some reassurance that the random numbers offered by computers are “random enough”, I’ve put a random number generator below for the skeptical reader to try. You can look at each number in its totality, or just follow certain digits – the fifth digit of each number produced, for instance. For random numbers between 1 and 100, you could follow the fifth and sixth digits, for another example.

This random number generator is powered by the Javascript built-in Math.random() function. Like many, it produces a number between 0 and 1. Users can customize the results, if desired, using other mathematical functions that multiply and round.

Your random number is  

I’ll be discussing Javascript and other aspects of tonight’s post in future ones:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl programming: random numbers

Tutoring math, you realize that some of your students will move on to computer science at university.  The tutor brings up a time-honoured favourite among comp-sci people:  random number generation.

Every computer programming language I know of has a random number generator.  The one I know in Perl is rand().  Common among such functions, it gives a random number between 0 and 1:  a decimal to 10 or so places.  If you want a random whole number, you can  multiply the output of rand() by 10, 100, 1000, or some other power of 10, then truncate it.  The following little program serves an example:

 
#!/usr/bin/perl

$num0=rand();

$num1=1000*$num0;

$num2=int($num1);

print “\n\nYour random number between 0 and 1000 is $num2\n\n”;

How are random numbers accomplished? There are several ways a computer might do so. A deeper question is, “How can a computer generate a truly random number, when a computer calculates numbers rather than creates them?” The answer lies in the compromise of accepting unpredictable as random.

For more about this wonderful topic, please return soon.

Source: perlmeme.org

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Ecology: environmental energy

Out for a walk in the woods recently, the tutor couldn’t help but arrive at this topic.  Like so many others, it’s relevant, yet rarely covered during tutoring….

On a hot day like we’ve had here lately, a hiker might notice the jungle-like foliage in the forest.  Then, feeling the sweat on their brow, and the insects on their bare arms, they might ask, “Is this what it’s like in the tropics?”

The answer is that, in the summer, conditions here can be similar to some much hotter places.  Of course, the difference is that they get those conditions the year round – or nearly year round.

Why is there so much more biological activity when it’s hot?  It comes down to the formula

KEave=3kT/2

where KE means kinetic energy, while T means temperature. The reality this equation points to is that the environment’s available energy increases with rising temperature.

Specifically, kinetic energy is the form of energy that is embodied in the motion of the molecules. The faster they move, the higher their kinetic energy. It is kinetic energy that facilitates chemical reactions, which is why turning up the heat on food cooks it faster.

When the outside temperature is higher, plants grow faster because growth is the sum of many chemical reactions. The plants, such as the grasses, feed everything else. In a tropical grassland, an acre has a much higher output of vegetation than it would have here on Canada’s west coast. Therefore, it can feed many more animals than an acre here. The result is not only more animals on the acre of tropical grassland, but a greater variety of them.

If it was always as warm here as it’s been lately, we would have a comparable biodiversity to a tropical setting. However, that might likely include poisonous snakes, parasites and diseases that Canadians never need think about – unless perhaps they travel to some tropical places.

Further exploration of the formula

KEave=3kT/2

and its energy implications will be explored in a future post:)

Source: hyperphysics.phy

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl programming: the user input array

The tutor notices that, given it’s Aug 14, we need a little progress in our summer project of Perl programming.  This rarely comes up in tutoring, but it’s a labour of love….

 
Back on June 22, I broached the idea of getting started with Perl. After all, I argued, academic pursuits can continue through the summer, the motivation being interest rather than preparing for an exam.

For those of you who actually took up the challenge and got started, you likely followed my blog through the summer. Not every article has been about Perl, but a good few have. The time has come for more.

Today’s article plugs an obvious hole in our knowledge base so far: getting user input. After all, it’s hard to interact with a program if you can’t give it different values to see how the output changes.

Perl has an array called @ARGV that is dedicated to storing inputs from the command line. (See my article on arrays here.) The following program serves as an example:


#!/usr/bin/perl
$name=$ARGV[0];
$firstnum=$ARGV[1];
$secondnum=$ARGV[2];
$ans=$firstnum + $secondnum;
print “\n\nHello, $name.”;
print ” You entered $firstnum and $secondnum.”;
print ” Their sum is $ans\n\n.”

The activity of this code is likely not mysterious. It is given the input array @ARGV with three values in it. $ARGV[0] is expected to contain your name; $ARGV[1] and $ARGV[2] should each contain a number. The program fetches your name from $ARGV[0], then stores it in the variable $name. Next, the program fetches the numbers from $ARGV[1] and $ARGV[2], stores them in $firstnumber and $secondnumber, adds the two, then stores the sum in $ans. On the command line, it gives you a friendly greeting by the name you gave, afterwards reminding you the numbers you gave and telling you their sum. Where you see \n, the terminal will start a new line. That’s just done to make space on the screen for the output.

At this point, the serious reader might have four questions in mind:

  1. How does your name get into $ARGV[0]?
  2. How does the first number get into $ARGV[1]?
  3. How does the second number get into $ARGV[2]?
  4. How does @ARGV get handed to the program as input?

The answer is that you enter those inputs, in order, after the program name. Let’s imagine the program above is called lucy.txt,and that your own name is Edward. Furthermore, you are just now wondering what the sum of 329 and 1982 is. Without a pencil or calculator handy, you’d like to call on lucy.txt for the answer. First, you will go into the terminal (see my articles here, here, and/or here, depending on your operating system). Next, you will place yourself in the directory where lucy.txt resides. Finally, you will call her and give her the three inputs by entering the following line:

perl lucy.txt Edward 329 1982

If all goes well, Lucy will answer with

Hello, Edward. You entered 329 and 1982. Their sum is 2311.

The very first line of lucy.txt, specifically

#!/usr/bin/perl

is called the shebang line. (You can read more about it here.) It is needed in non-Windows environments. While it’s not necessary in Windows, it doesn’t seem to hurt if it’s there.

Good luck with this!

Source: Robert Pepper’s Perl Tutorial

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.