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Math Help - Divide a line in to multiple section of increasing length according to a ratio

  1. #1
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    Divide a line in to multiple section of increasing length according to a ratio

    hi,

    I want to divide a line segment which is of a fixed length, in to multiple section, where each successive section is greater than preceding section in length by a particular factor 'a' where 'a > 1'.
    There is a also a minimum and maximum limit on the length of the sections.

    for example if I have a line segment of length 10, and I want to divide it in to sections
    where minimum section length is 1 and maximum section length is 3.

    so my question is that is there any way using which I can determine the number of sections 'n' which can probably give a close result to divide a the line with minimum length approximately equal to 1 and maximum length 3, and each section other than the first one is greater than the previous section by a factor a.
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  2. #2
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    Hello gourish

    Welcome to Math Help Forum!
    Quote Originally Posted by gourish View Post
    hi,

    I want to divide a line segment which is of a fixed length, in to multiple section, where each successive section is greater than preceding section in length by a particular factor 'a' where 'a > 1'.
    There is a also a minimum and maximum limit on the length of the sections.

    for example if I have a line segment of length 10, and I want to divide it in to sections
    where minimum section length is 1 and maximum section length is 3.

    so my question is that is there any way using which I can determine the number of sections 'n' which can probably give a close result to divide a the line with minimum length approximately equal to 1 and maximum length 3, and each section other than the first one is greater than the previous section by a factor a.
    There's no way to get exactly the answers you need - unless you just happen to be lucky.

    If the minimum section is of length \displaystyle p, the maximum \displaystyle q, the common ratio between consecutive sections \displaystyle a, the total length \displaystyle S and the number of sections is \displaystyle n, then the lengths form a geometric progression. This gives:
    \displaystyle q = pa^{n-1}
    and
    \displaystyle S = \frac{p(a^n-1)}{a-1}

    Since \displaystyle n has to be an integer, a solution to this pair of simultaneous equations is unlikely to be possible. A spreadsheet will give you some approximate answers. In the case you quote, \displaystyle p=1,\; S=10 we get:
    When \displaystyle n = 5 and \displaystyle a=1.352, \displaystyle q\approx3.345
    and
    When \displaystyle n = 6 and \displaystyle a=1.203, \displaystyle q\approx2.517

    Grandad
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