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Math Help - Work out length of arc

  1. #1
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    Question Work out length of arc

    Hi, i am studying maths on khan academy and got stuck with a certain problem and i can't find any videos which explain it so i could do with some help please.

    When i look up how to do this problem on the internet it says i just do 321/360 * the circumference. simple but i want to know how to work it out the way they have done it in the image. I don't understand how he gets from step 2 to step 3, how he gets the 321/360 to 107/6. so the question isn't really geometry but i didn't know where else to put it.


    "A circle with circumference 20π has an arc with a 321 central angle. What is the length of the arc?"

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  2. #2
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    Re: Work out length of arc

    The angle tells you the proportion of the circle. So it's just \displaystyle \frac{321}{360} \times 20\pi .
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  3. #3
    MHF Contributor ebaines's Avatar
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    Re: Work out length of arc

    It's simple algebra:

     \frac {321}{360} = \frac s {20 \pi}

    Rearrange:

     s = 20 \pi \frac {321}{360} =  \pi \frac {321}{ 18} = \pi \frac {107}{6}
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    Re: Work out length of arc

    ebaines yes that's it, but it's the second step of yours that i always get stuck on. how did you know to divide the 360 by 20?

    how do i get used to using tricks like this? is it algebra i need to study more?
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  5. #5
    MHF Contributor ebaines's Avatar
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    Re: Work out length of arc

    Simplifying fractions involves looking for common factors in both the numerator and denominator and canceling them out. So for example:

     \frac 6 8 = \frac {2 \times 3}{2 \times 4} = \frac  3 4

    Don't be confused by the fact that in my previous post I showed the  20 \pi factor in front of the fraction as opposed to being in the numerator:

     20 \pi \frac {321}{360} = \frac {20 \times  \pi \times 321}{20 \times 18 } = \frac {\pi \times 321}{18} = \frac {\pi \times 3 \times 107}{3 \times 6} = \frac {107  \pi}{6}
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    Re: Work out length of arc

    \frac{ac}{ab}= \frac{c}{b} so \frac{(20)(321)}{360}= \frac{(20)(321)}{(20)(18)}= \frac{321}{18}

    Of course, you don't have to do it that way. You can go ahead and multiply to get \frac{(20)(321)}{360}= \frac{6420}{360}
    Then it should be obvious that both are divisible by 10: \frac{6420}{360}= \frac{642}{36}
    And, since both of those are even, they are divisible by 2: \frac{642}{36}= \frac{321}{18}
    It is not quite so obvious but those are both divisible by 3: \frac{107}{6}.

    That's not algebra- that's arithmetic.
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    Re: Work out length of arc

    That really helped, thanks. You're right ebaines it was the way it was positioned in front of the fraction that confused me. All is clear now and i can progress.
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    Re: Work out length of arc

    Does anyone know any good websites or books that can help you understand what you can move around in equations? think i need some practice with this.
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