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anonymous

  • one year ago

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  1. anonymous
    • one year ago
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    The arcs in the photo (will give later on below) to the right appear to be paths of stars rotating about the North Star. To produce this effect, the photographer set a camera on a tripod and left the shutter open for a long time. If the photographer left the shutter open for a full 24 hours, each arc would be a complete circle. You can model a star’s“rotation” in a coordinate plane. Place the North Star at the origin.Let P(1, 0) be the position of the star at the moment the camera’s shutter opens.Suppose the shutter is left open for 2 hours and 40 min.(1)Find the angle of rotation that maps point P on to P’ 2. (2)What are the x and y coordinates of point P’ to the nearest thousandths? (3) Determine a translation rule that maps point P onto P

  2. anonymous
    • one year ago
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  3. anonymous
    • one year ago
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    2 hr 40 min = \(2\frac{ 2 }{ 3 }\) hr In 24 hrs the rotation is 2π radians. Set up a proportion to find the angle of rotation between the P and P'. \[\frac{ 24 }{ 2\pi }=\frac{ 2\frac{ 2 }{ 3 } }{ \theta }\]

  4. anonymous
    • one year ago
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    Then you can use trig ratios to find the x and y coordinates of P' since the radius of the circle = the hypotenuse of the right triangle = 1

  5. anonymous
    • one year ago
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    @peachpi can you explain more? I still don't understand

  6. anonymous
    • one year ago
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    slight mistake \[rate = \frac{ distance }{ time }\] The rate of the rotation is the same for the whole 24 hours. It takes 24 hours to rotate 2π radians (or now that I think about it you probably want this in degrees, so 360°). We're trying to find the angle that corresponds to the time of \(2\frac{ 2 }{ 3 }\) hours. \[\frac{ 360° }{ 24~hours }=\frac{ \theta }{ 2\frac{ 2 }{ 3 }~hours}\] or if you wanted radians \[\frac{ 2pi }{ 24~hours }=\frac{ \theta }{ 2\frac{ 2 }{ 3 }~hours}\]

  7. anonymous
    • one year ago
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    hummmm, so this is the rotation from point P to P'? @peachpi

  8. anonymous
    • one year ago
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    Θ is the angle of rotation

  9. anonymous
    • one year ago
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    @peachpi there is 2 more question go with the proplem, can you slove it too?

  10. anonymous
    • one year ago
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    |dw:1439692128942:dw|

  11. anonymous
    • one year ago
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    Use what you get for x and y for the translation. P is at (1, 0). P' is at (x, y), Once you have x and y find the numbers you need to add/subtract to turn (1, 0) into (x, y)

  12. anonymous
    • one year ago
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    okay, thanks

  13. anonymous
    • one year ago
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    @peachpi Since I got the results is x = 0.643, y=0.776, I know the translation rule so I was just wondering if this right or not: T(x,y) = (x+0.643,y+0.776)

  14. anonymous
    • one year ago
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    The y would be right. For the x, you want to turn 1 into 0.643, so you'd have to subtract 0.357. So it's T(x, y) = (x - 0.357, y + 0.776)

  15. anonymous
    • one year ago
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    okay :) thanks:)

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