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Math Help - arc length

  1. #1
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    arc length

    The figure shows the part of a circle whose circumference is 45. If arcs of length 2 and length b continue to alternate around the entire circle so that there are 18 arcs of each lenth, what is the degree measure of each of the arcs of length b?

    ans is 4 degree, but I don't know why...
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  2. #2
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    Quote Originally Posted by Judi
    The figure shows the part of a circle whose circumference is 45. If arcs of length 2 and length b continue to alternate around the entire circle so that there are 18 arcs of each lenth, what is the degree measure of each of the arcs of length b?

    ans is 4 degree, but I don't know why...
    Total length of 18 arcs of length 2 is 36,
    Total length of 18 arcs of length b is 18b.

    These two total 45, so:

    36+18b=45,

    or:

    18b=9.

    So b=1/2.

    RonL
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  3. #3
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    From 2{\pi}r=45, the radius of the circle is \frac{45}{2{\pi}}

    Using s=r{\theta}, we find {\theta}=\frac{2}{\frac{45}{2{\pi}}}=\frac{4{\pi}}  {45}, the central angle subtended by 1 arc of length 2.

    Multiply this by 18 and that's the total central angle of the circle(in radians) subtended by the arcs of length 2.

    (18)\frac{4{\pi}}{45}=\frac{8{\pi}}{5}

    There are 2{\pi} radians in a circle, so subtract and that will tell you how much of the circle is left, which is the central angle of the circle subtended by the 18 b's.

    2{\pi}-\frac{8{\pi}}{5}=\frac{2{\pi}}{5}

    Divide by 18 and that's how much central angle is subtended by each b.

    \frac{\frac{2{\pi}}{5}}{18}=\frac{\pi}{45}

    Convert to degrees:

    (\frac{\pi}{45})(\frac{180}{\pi})=\Huge{4} degrees

    There's numerous ways to go about this, but I wanted to show you the circle properties and how they work. Let me know if you don't understand something. Okey-doke?.
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