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Thread: Exponential loss of light

  1. #1
    Oct 2010

    Exponential loss of light

    Here is a problem I'm having trouble with:

    50 8.0
    3. Underwater light intensity decreases with increasing depth: the IDTRMn
    deeper you go, the darker it gets. The relationship is approximately
    an exponential attenuation, such that

    A =B exp(-Kz)

    where A is light intensity at depth z, B is light intensity at
    depth 0, and K is the attenuation coefficient. The table below
    lists depth z on the left and measurements of A on the right. Find K
    A when z = 0 (which is the same thing as B).

    2 1172.2
    4 963.1
    6 737.1
    8 636.6
    10 512.9
    12 438.2
    14 341.3
    16 284.6
    18 223.1
    20 176.1
    22 158.0
    24 115.7
    26 84.8
    28 85.7
    30 67.9
    32 54.4
    34 42.5
    36 36.0
    38 28.7
    40 22.1
    42 18.0
    44 14.7
    46 12.0
    48 9.8
    50 8.0


    Alright, so what I did to try and solve this was set up a system of equations. I solved for B at a depth of 10, and plugged in that equation for the constant B itself in a new depth equation for depth 12. This gave me an equation of 438.2 = (512.9/(e^-K(10)))*e^-K(12). I solved for K, got a number, then plugged that value back into the equation and solved for B.

    Now the values I got using this method for B and K both worked for each of these equations. However, when I tried using the same values for different depths, the answers got further and further away from the actual values. I guessed that solving for two depths next to each other in the table was giving me a poor approximation, so what I did was I used the same method for depth 2 and depth 50. This gave me a better approximation, but I'm still off by a fair number the furthest away from either values for 2 or 50 (depth 30 is 67.9 but I got something like 64 or 63).

    I could probably use the latter method here to get a relatively good answer for depth zero, but this is just bugging me. How do I solve this so that I can get the values of K and B to be almost exact algebraically? I might be able to plug this into a spreadsheet and do it that way, but that's not exactly showing my work or explaining how the answer can be obtained.
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  2. #2
    MHF Contributor
    Oct 2009
    My guess is that you are taken advantage of Probably the teacher's cousin, who is a marine biologist, got this numbers from an actual experiment and thought, "Why don't I give this to the kids in my cousin's math class; let's see what they can figure out".

    Seriously, I think what you did is right and the problem here is that data fluctuates a lot. Note, for example, that the intensity at 28 is greater that at 26. Also, the ratio of intensities at 20 and 22 is 1.11 while the ratio between 22 and 24 is 1.37.

    I did plug the numbers in a spreadsheet and calculated K and B based on every pair of consecutive measurements. The standard deviation of B turned out to be 1212.0, while the average was 1854.3. This shows how wild the data is.

    The best-fitting K and B computed by the LOGTEST function (OpenOffice 3.2) were -0,104 and 1458.159, respectively. OpenOffice uses linear regression (the "least squares" method). The average of K's and the median of B's computed from each pair were 0,1039
    1468,741, which is pretty close.

    Your solution probably should depend on what the question is testing. If you were studying the least squares method, then you should apply it. If you only studied the exponential function, then what you did is probably sufficient.

    Disclaimer: I am not a statistician. You may get a more authoritative answer in one of the statistics forums on this site.
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  3. #3
    Oct 2010
    thanks emakarov. the question doesn't specifically ask for standard deviation, but this guy might dock me points for not having it. that said, do you have/could you make the .ods file with your work in a spreadsheet and attach it here? I'd like to take a look at how you set it up, so I know how it got all worked through and I can write down that the data is all over the place, and this is how i can prove it.


    actually scratch that, i just did it myself and obtained the exact same things you did. thanks for the info, i can now fully answer the question (and fully rip apart the prof for giving us such numbers).
    Last edited by shrinkshooter; Oct 4th 2010 at 11:18 AM.
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