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Estimation example from Giancoli

  1. Aug 27, 2015 #1


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    Hello, I´m a bit math-challenged and just started to brush up on both physics and mathematics simultaneously but I'm still at absolute beginner status and I don't want to go on in the book as long as I can't even figure out such a simple example. I'm using the Giancoli textbook on physics for self-study but can't quite follow the simplest of examples in the first chapter. Since the problem lies in the mathematical part of my understanding and not the conceptually physical part I thought I'd post it here. Just wanted to check if I'm doing something wrong or whether there's a typo in the example because unfortunately there doesn't seem to be a list of errata for the book. Here it is:

    1. The problem statement, all variables and given/known data

    The example is about estimating the volume of a lake by using some rough measures and rounded-off numbers to get at the approximate order-of-magnitude.
    So it says:

    V=hπr2 ≈(10meters) × (3) ×(5×102meters)2≈8×106m3 ≈ 107 m3

    where π has been rounded to 3, the average depth of the lake is 10m and the estimated radius of the surface is given by 5×102 (500 m).
    I can't seem to follow how he arrives at 8 ×106.
    2. Relevant equations
    3. The attempt at a solution
    In my mind it would come to 3×(5×105) and then maybe 1.5×101 ×105 and then 1.5 ×106. This is of course different from the given 8×106.

    I figure I'm probably making a mistake here in the rules of multiplication of exponents and bracketing out but I can't figure out what it is. Any help would be much appreciated. Sorry for the absolute rookie level of mathematical (in)competence.
  2. jcsd
  3. Aug 27, 2015 #2


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    Don't forget to square the radius: (5×102)2 = 500 × 500 = 250,000

    Multiply by the average depth: 250,000 × 10 = 2,500,000

    Multiply by π (approx.): 2,500,000 × 3 = 7,500,000

    Round final result to 1 significant figure: 7,500,000 ≈ 8,000,000 or 8×106 m3

    Personally, I find writing some numbers in scientific notation tedious and confusing (for example, 5×102 for 500).

    When you do square numbers written in scientific notation, remember to square the first part but double the exponent on the 10:

    (5×102)2 = 25×104 = 2.5×105 = 250,000
  4. Aug 27, 2015 #3


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    Ah right of course. Thanks a million! That made a lot of sense and really helped me!
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