Question

Will give medal and fan :) thank you
question and photos below

Answer 1

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

Answer 2

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'.
\[\frac{ 24 }{ 2\pi }=\frac{ 2\frac{ 2 }{ 3 } }{ \theta }\]

Answer 3

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

Answer 4

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

Answer 5

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
\[\frac{ 2pi }{ 24~hours }=\frac{ \theta }{ 2\frac{ 2 }{ 3 }~hours}\]

Answer 6

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

Answer 7

Θ is the angle of rotation

Answer 8

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