Monday, August 6, 2012

Beauty by Numbers

As I walk the shores of Lake Huron, I am put in mind of Oliver Wendell Holmes's The Chambered Nautilus, which please join me in taking as a natural (ha!) link from last week's post to this week's exploration of Fibonacci.  Having read these two pieces, consider the following, adapted from Arup Guha's links to his TOK past:

Most math problems have multiple solutions that are significantly different. In this assignment, choose a single math problem from the 4 below and solve it two different ways, the more different, the better. After showing each solution to your problem, discuss which of the two solutions you feel is more elegant/beautiful.

An example problem is shown here:

Find the area of a triangle with side lengths 5, 12, and 13:

1)    Solve for the angle across the side 13 using the law of cosines:

cos C = (52 + 122 – 132)/(2*5*12)
cos C = 0
C = 90°, thus, 5 and 12 form a base and height for the triangle and the
area is (.5)(5)(12) = 30.

2)    Use Heron’s formula: A = √(s)(s-a)(s-b)(s-c), where s is the semiperimeter of the triangle.

S = (5+12+13)/2 = 15
Area =  √(15)(15-5)(15-12)(15-13) = √(15)(10)(3)(2) = √900 = 30

Please choose one of the following problems for your posting:

1) If log(xy3) = 1 and log(x2y) = 1, what is log (xy)?

2) Mr. Earl E. Bird leaves his house for work at exactly 8:00A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60 miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?

3) What is the area of the triangle bounded by the lines y = x, y = -x and y = 6?

4) Given that x2+y2 = 14x + 6y + 6, what is the largest possible value that 3x+4y can have?


  1. I chose to answer question #1 even though logarithms have proven to be a point of weakness for me.

    1.)If log((xy^3) = 1 and log(x^2y) = 1, then log(xy^3) = log(x^2y)

    From that equation, x is divided from both sides leaving log(y^3)= log(xy)

    2.)If log((xy^3) = 1 and log(x^2y) = 1, then log(xy^3) = log(x^2y)
    The above equation can be rewritten, because of the laws of logarithms as:
    log(x)+ 3log(y)= 2log(x)+ log(y)

    log(x) can be subtracted from each side and then the equation can be simplified to finalize with:
    log(y^3)= log(xy)

    I prefer the second of the two solutions because he answer is easier to see though the steps prior to the answer are harder. One must know the laws of logarithms in order to break the problem down to be simplified.

  2. 1) If log(xy3) = 1 and log(x2y) = 1, what is log (xy)?
    log (xy) = 1
    x and y have to equal 1 for the answer to always be 1.

    I don't know of another way to solve this problem.

  3. 1) If log(xy3) = 1 and log(x2y) = 1, what is log (xy)?

    I had the same thinking as Lola (above). I know from Ms. LT that both have to be set equal to each other and if work needed to be shown, I would have done the same as in solution number two for Kate, but if I had to do it by just thinking, I would have done the same as Lola because of the rules for logs.

  4. I chose question 2, because I remembered doing something similar in Math SL.
    So, i will start by making the number of hours travelled the unknown, thus X. After that, i will use a calculator to convert the minutes in hours. Therefore 3 minutes=0.05hours.
    40(X+0.05)=60(X-0.05). So then after, I will have to distribute the 40 and the 60 in the parenthesis. Which will give me: 40X+2=60X-3, So -20X=-5, So X=0.25
    Now I can plug back in the unknown (x) in the equation, 40(0.25+0.05)=12. Thus, from that I can find his average speed by doing 12/0.25=48miles per hour.
    I don't think we have been taught of another way to solve this problem, but one can still try the guess and check method, but i fear they would have to guess and check for a very long time. Thus, the way I did it is more effective or efficient.

  5. 1)log(xy^3)= log(x^2y) 2)log(x)+ log (y^3)= log(x^2)+ log (y)
    /x /log(x)
    log(xy)= log(y^3) log(y^3)= log(x)+log(y)

    For the logarithm problem, I did them almost like my good friend Kate. I find beauty in simplicity, so I prefer the first problem. Since the post asks or which one is elegant/ beautiful, I must say that the second way was much more elegant than the first way. Dragging the problem out from a compact equation to a much longer, thinned out equation really is elegant.

  6. I choose to do #4, and I choose to use my graphing calculator to solve the question, and this is a video from youtube which uses the same way as how I did:
    except that, he did the problem two different ways and I do think that the second way of solving the problem is more precise and fast, and therefore I think it is a more "elegant" way to solve the problem.

  7. I chose question #1 as well, because I'm friends with logs.
    1) If log(xy3) = 1 and log(x2y) = 1, what is log (xy)?
    Like Sara and Lola, I have the same thinking, that x=1 and y=1, because that makes each log equal to one. When looking back at the log properties for part 1 of our summer math homework. Also, I always like to check with my graphing calculator by graphing the equations to check that they are equal to each other. But since this one was simple enough for me to do in my head.

  8. I chose to do question #2 and it took me a while to come up with a second way to answer it. After reading the question, I automatically said 50 miles per hour since 3 minutes is the same difference when he is both late and early with, with the average speed of 60 miles per hour and 40 miles per hour. I figured the answer has to the average of both numbers. To find the average, first I drew a number line, and the number with the exact difference of numbers between 40 and 60 is 50. My second way was to add 60 and 40 and divide the answer, 100 by 2 and get 50.
    I personally prefer the second one because it is quicker and I do not have to draw a number line. Though I guess, the most elegant would be the first way I answered the question because it gives a visual for people to see and comprehend and it shows another part of math that is not just numbers.

  9. I chose to do #1 as well:

    Like my classmates I knew that if (xy^3)=1and log(x^2y)=1 then (xy^3)=(log(x^2y).
    So by dividing the x on both sides it should equal (xy)=1.
    You could also just do this in your head. You could assume that because (xy^3)=log(x^2y) and therefore (xy)=1 as well.

    But I would have to say that the first method is the most elegant because the long process of the numbers creates a beautiful pattern.


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