#$&*
course Mth 164
9/21/2011 around 4 pm
If your solution to stated problem does not match the given solution, you should self-critique per instructions athttp://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
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Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
001. Radian measure and the unit circle.
Goals for this Assignment include but are not limited to the following:
1. Know the definition of the radian.
2. Relate coordinate positions on the unit circle to angular displacement and to arc displacement.
Click once more on Next Question/Answer for a note on Previous Assignments.
Previous Assignments:
Be sure you have completed all Preliminary Assignments as instructed on under the Assts link on the homepage. These assignments include the q_a_orientation, and the three sets Initial Problems, Describing Graphs and Typewriter Notation from the q_a_init_pbs program. Links and explanations are included on the Assignments Page.
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Question: `q001. Note that there are 10 activities in this assignment.
Figure 37 (located under the Figures link on the Assignments page under Assignment 0) depicts a circle of radius 1 centered at the origin of a x y coordinate system. Imagine the we have 2 ants, one red and one black. Both start out moving at the same speed from the point for the positive x-axis beats the circle. The red ant crawls along the arc of the circle in the counterclockwise direction, and black ant crawls along the x-axis toward the origin. The ants proceed until the black ant reaches the origin. Both ants will have crawled the same distance, the black ant along a straight line and the red ant along an arc of the circle.
At that instant the red ant will have traveled a distance equal to 1 radius of the circle, and we say that the red ant has completed 1 radian of arc. Which of the indicated points on the circle will correspond to a 1 radian arc? Note that we have indicated points a, b, c, d.
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Your solution:
Point B would be the best answer since point a is too close to the initial point and points c and d are too far away to be considered.
confidence rating #$&*: 2
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Given Solution:
`aWe see visually that the point a lies at an arc distance less than the radius of the circle. We also see that the point c lies at an arc distance that is clearly greater than the radius of the circle. The only possible candidate for a 1 radian angle, which must lie at an arc distance equal to one radius, is therefore point b.
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Self-critique (if necessary):
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Self-critique Rating:OK
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Question: `q002. If the first ant moves at a constant speed, moving through 1 radian every second, then approximately how long, to the nearest second, do you think it will take for the ant to move along the arc to the point where the circle meets the negative x-axis?
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Your solution:
There are 3 points to get to the Point c on the negative x-axis, therefore if it takes 1 radian a second, it would take 3 seconds to get to point c, 1 radian per second times 3 radians.
confidence rating #$&*: 3
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Given Solution:
`aVisual examination, perhaps accompanied by a quick sketch, shows that it takes approximately 3 arcs each of one radian to get from the positive x-axis to the negative x-axis when moving along the arc of the circle.
In figure 37 the points b, c and d lie at approximately 1, 2 and 3 radians. Remember that each radian corresponds to an arc distance equal to the radius of the circle.
At 1 radian / second it will take about 3 seconds to move the approximately 3 radians to the negative x axis.
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Self-critique (if necessary):
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Self-critique Rating: OK
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Question: `q003. If the ant traveled at 1/2 radian per second, then after 1 second would its angular position be indicated by point a, point b, point c or point d in Figure 37
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Your solution:
If the ant traveled at ½ radian per second then its angular position would be point a since it takes 1 radian per second to get to the point a.
confidence rating #$&*: 3
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Given Solution:
`aAfter 1 second the angular position would be 1/2 radian, which would correspond to point a.
Note that after 2 seconds the angular position would be 1 radian, corresponding to point b, and after three seconds the angular position would be 3 * 1/2 radian = 3/2 radian and the ant would be at position c.
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Self-critique (if necessary): I was corresponding the last question to this one with the seconds. I understand how to get radians per second and I got the question right, but I don’t think I answered it exactly how it should have been worded.
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Self-critique Rating: OK
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Question: `q004. How far will the ant travel in the process of completing 1 trip around the circle, starting and ending at the initial point where the circle meets the positive x-axis.
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Your solution:
If the ant traveled around the circle one time, that would be the circumference of the circle which is a radius of 6.28 or 2 pi r.
confidence rating #$&*: 3
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Given Solution:
`aThe circumference of the circle is 2 pi r, where r is the radius of the circle. This is the distance traveled by the ant.
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Self-critique (if necessary):
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Self-critique Rating: OK
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Question: `q005. As we just saw the distance around the circle is its circumference 2 pi r, where r is the radius. Through how many radians would the ant travel from the initial point, where the circle meets the positive x-axis, if the motion was in the counterclockwise direction and ended at the original point after having completed one trip around the circle.
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Your solution:
This answer would be similar to the last question, except in the counter-clockwise process. It would still be 2 pi radians. Each radian equals 2 pi which would equal approximately 6.28 radians around the circle.
@& Each time you move a distance along the arc which is equal to the radius, that's another radian.
You move 2 pi times the radius as you go around once, so that's 2 pi radians.*@
@& Question: `q006. The unit circle is a circle of radius 1 centered at the origin. What are the coordinates of the points where the unit circle meets the positive x-axis, the positive y axis, the negative x-axis and the negative y axis?*********************************************Self-critique Rating: ok------------------------------------------------Self-critique (if necessary): I don’t think I answered this question exactly correct either. I know that a complete circle is 2 pi r and that each segment is one radian of arc so there are fractions for each radian.&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&The arc distance around the circle is 2 pi r, which therefore corresponds to 2 pi r / r = 2 pi radians.We understand by these examples that if we divide the arc distance by the radius, we will get the number of radians of angular distance.Question: `q005. As we just saw the distance around the circle is its circumference 2 pi r, where r is the radius. Through how many radians would the ant travel from the initial point, where the circle meets the positive x-axis, if the motion was in the counterclockwise direction and ended at the original point after having completed one trip around the circle. *********************************************Self-critique Rating: OK------------------------------------------------Self-critique (if necessary):&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&`aThe circumference of the circle is 2 pi r, where r is the radius of the circle. This is the distance traveled by the ant.Given Solution: Question: `q004. How far will the ant travel in the process of completing 1 trip around the circle, starting and ending at the initial point where the circle meets the positive x-axis.*********************************************Self-critique Rating: OK------------------------------------------------Self-critique (if necessary): I was corresponding the last question to this one with the seconds. I understand how to get radians per second and I got the question right, but I don’t think I answered it exactly how it should have been worded.&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&Note that after 2 seconds the angular position would be 1 radian, corresponding to point b, and after three seconds the angular position would be 3 * 1/2 radian = 3/2 radian and the ant would be at position c.`aAfter 1 second the angular position would be 1/2 radian, which would correspond to point a. Question: `q003. If the ant traveled at 1/2 radian per second, then after 1 second would its angular position be indicated by point a, point b, point c or point d in Figure 37 *********************************************Self-critique Rating: OK------------------------------------------------Self-critique (if necessary):&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& At 1 radian / second it will take about 3 seconds to move the approximately 3 radians to the negative x axis.Question: `q002. If the first ant moves at a constant speed, moving through 1 radian every second, then approximately how long, to the nearest second, do you think it will take for the ant to move along the arc to the point where the circle meets the negative x-axis?*********************************************Self-critique Rating:OK------------------------------------------------Self-critique (if necessary):&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&`aWe see visually that the point a lies at an arc distance less than the radius of the circle. We also see that the point c lies at an arc distance that is clearly greater than the radius of the circle. The only possible candidate for a 1 radian angle, which must lie at an arc distance equal to one radius, is therefore point b.Given Solution: self-critique
@& The line breaks were missing from the second half of your document. I think I was able to locate all your responses, but just in case, you're welcome to submit another copy which includes all the line breaks. I don't mind looking again.*@
@& Do see my comment(s).*@