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Math Help - Word problem about alternating series (I think)

  1. #1
    s3a
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    Word problem about alternating series (I think)

    Here is the question:
    "Two boys on bicycles, 35 miles apart, began racing directly toward each other. The instant they started, a fly on the handle bar of one bicycle started flying straight toward the other cyclist. As soon as it reached the other handle bar it turned and started back. The fly flew back and forth in this way, from handle bar to handle bar, until the two bicycles met. If each bicycle had a constant speed of 13 miles an hour, and the fly flew at a constant speed of 21 miles an hour, how far did the fly fly?"

    Intuitively, (I think) I see that this is describing an alternating series since the two bikes will be closer and closer and therefore the sum of the distances the fly has to travel back and forth will converge (conditionally) to a point. I just don't know how to actually solve the problem numerically and would greatly appreciate it if someone could show me how.

    Thanks in advance!
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    Re: Word problem about alternating series (I think)

    Quote Originally Posted by s3a View Post
    Here is the question:
    "Two boys on bicycles, 35 miles apart, began racing directly toward each other. The instant they started, a fly on the handle bar of one bicycle started flying straight toward the other cyclist. As soon as it reached the other handle bar it turned and started back. The fly flew back and forth in this way, from handle bar to handle bar, until the two bicycles met. If each bicycle had a constant speed of 13 miles an hour, and the fly flew at a constant speed of 21 miles an hour, how far did the fly fly?"

    Intuitively, (I think) I see that this is describing an alternating series since the two bikes will be closer and closer and therefore the sum of the distances the fly has to travel back and forth will converge (conditionally) to a point. I just don't know how to actually solve the problem numerically and would greatly appreciate it if someone could show me how.

    Thanks in advance!
    there is a much easier solution to this ... note that (rate)(time) = distance
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  3. #3
    s3a
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    Re: Word problem about alternating series (I think)

    Yes, so, I was thinking. The combined relative speeds of one of the boys and the fly is 13+21 = 34 miles an hour so at t = (35/34)h, the fly landed on the first handle bar (that is not the one it started with). And then, I compute the distance traveled by the boys, take a mental snapshot there and again, using the relative speed of the other boy and the fly which is still 34 miles an hour, I find where the fly will intersect with a boy again and then I figure out the general style of the series.

    But again, I still keep getting it wrong so please help me.
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    Re: Word problem about alternating series (I think)

    Quote Originally Posted by s3a View Post
    Yes, so, I was thinking. The combined relative speeds of one of the boys and the fly is 13+21 = 34 miles an hour so at t = (35/34)h, the fly landed on the first handle bar (that is not the one it started with). And then, I compute the distance traveled by the boys, take a mental snapshot there and again, using the relative speed of the other boy and the fly which is still 34 miles an hour, I find where the fly will intersect with a boy again and then I figure out the general style of the series.

    But again, I still keep getting it wrong so please help me.
    how much time does it take for the two bicycles to meet?
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    s3a
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    Re: Word problem about alternating series (I think)

    (35/26) hours
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    Re: Word problem about alternating series (I think)

    Quote Originally Posted by s3a View Post
    (35/26) hours
    isn't the fly traveling at 21 mph for that time also?
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  7. #7
    s3a
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    Re: Word problem about alternating series (I think)

    Yes. (Was that asking me something implicitly?)
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    Re: Word problem about alternating series (I think)

    Quote Originally Posted by s3a View Post
    Yes. (Was that asking me something implicitly?)
    ... take another look at my first response.
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    s3a
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    Re: Word problem about alternating series (I think)

    If you're suggesting: (35/26)h * 21miles/h, that doesn't make sense to me since first of all there is no alternation going on and, the fly could reach a boy before that boy reaches the other boy. Also, why can't you just show me how to do it? That way, I could've understood this a long time ago. (I'm not one of those people that quickly forget what was said if it's told to them - I pedantically analyze what I am given).
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    Re: Word problem about alternating series (I think)

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    s3a
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    Re: Word problem about alternating series (I think)

    Actually, I am still having trouble but I am at least better off than I use to be I think. Could you please go over my work and tell me what I did wrong? My work is attached as a pdf file.
    Attached Files Attached Files
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    s3a
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    Re: Word problem about alternating series (I think)

    I got it!

    If anybody cares, this is how to do it:

    13t = -13t + 35, t=35/26, d_fly = 21(35/26)
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