## anonymous one year ago Will give medal and fan :) thank you question and photos below

1. anonymous

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

3. anonymous

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

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

@peachpi can you explain more? I still don't understand

6. anonymous

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

hummmm, so this is the rotation from point P to P'? @peachpi

8. anonymous

Θ is the angle of rotation

9. anonymous

@peachpi there is 2 more question go with the proplem, can you slove it too?

10. anonymous

|dw:1439692128942:dw|

11. anonymous

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

okay, thanks

13. anonymous

@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

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

okay :) thanks:)