Assignment 1 QA

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course Mth 164

11/29 Am1/13/15

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:

I’m unsure about this solution but to me it seems that 1 radian arc is equal to the length from the initial point to point b.

It seems to me that this length is closer to the radius length than any of the other points.

confidence rating #$&*: 1

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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):

I was surprised that I correctly answered the question but the statement that it was equal to the radius made it seem simple to which of the points it would be.

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Self-critique Rating: 3

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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:

Looking at the image presented I think this would take a little over 3 seconds if the ant is traveling at 1 radian per second.

confidence rating #$&*: 1

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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):

OK

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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:

The ant traveling at ½ radian per second would travel from the initial point to point a after 1 second.

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 halfway between position b (which is at 1 radian) and position c (which is at 2 radians).

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Self-critique (if necessary):

OK

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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:

Looking at the circle I would think a round trip would take about 7 radians.

confidence rating #$&*: 1

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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):

I didn’t think to calculate the circumference for this problem and this is certainly reasonable for calculating the distance the ant travels.

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Self-critique Rating: 3

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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:

As I stated in the last problem I believe this would be about 7 radians.

confidence rating #$&*: 2

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Given Solution:

`aAn arc displacement of r corresponds to an arc distance of 1 radian on the circle. Arc distances of 2, 3, 4, ... time the radius would correspond to 2, 3, 4, ... radians of arc. That is, arc distance of r, 2r, 3r, 4r, ... correspond to 1, 2, 3, 4, ... radians of arc.

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.

The arc distance around the circle is 2 pi r, which therefore corresponds to 2 pi r / r = 2 pi radians.

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Self-critique (if necessary):

I think I understand we divide the circumference by the radius and this results in 2 pi radians. I’m a little unclear about the solution though.

???Would 2 pi radians be the number of radians to make a complete trip around the circle???

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Self-critique Rating: 1

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Yes.

A radian of angle corresponds to an arc on the circle whose length is equal to the radius.

Since the circumference is 2 pi times the radius, the corresponding angle is 2 pi radians.

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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?

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Your solution:

If I understand correctly the radius is 1 so the circle crosses the following at each coordinate:

Positive x axis at (1, 0)

Negative x axis at (-1, 0)

Positive y axis at (0, 1)

Negative y axis at (0, -1)

confidence rating #$&*: 2

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Given Solution:

`aThe unit circle has radius 1 and is centered at the origin, so the circle meets the positive x-axis 1 unit from the origin at (x, y) = (1,0). Similarly the circle meets the positive y-axis at the 'top' of the circle, 1 unit from the origin at (x, y) = (0,1); the circle meets the negative x-axis at (-1, 0); and the circle meets the negative y-axis at (0,-1).

Figure 84 shows these points on the unit circle. Note that in this figure the small dots are located at increments of .1 unit in the x and y directions.

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

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Question: `q007. Without looking at Figure 84, sketch a picture of the unit circle, complete with labeled points where the circle meets the x and y axes. Indicate the arc from the standard initial point, where the circle meets the positive x-axis, to the point where the circle meets the positive y axis. Describe your sketch.

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Your solution:

My sketch was smaller in scale to the presented picture so to save paper but I still used the same overall measurements.

My circle has a radius of 1 and meets each axes as follows:

Positive x at (1, 0)

Positive y at (0, 1)

Negative x at (-1, 0)

Negative y at (0, -1)

confidence rating #$&*: 2

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Given Solution:

`aYour sketch should show the x and y axes and a circle of radius 1, with the points (1,0), (0, 1), (-1, 0) and (0, -1) where the circle meets the coordinate axes labeled. The arc will run along the first quadrant of the circle from (1,0) to (0,1). Your figure should match figure 84.

You should be able to quickly draw this picture any time you need it.

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

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Question: `q008. How many radians of angular displacement correspond to the arc displacement from the standard initial point, where the circle meets the x-axis, to the point where the circle meets the positive y axis?

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Your solution:

I’m not sure I understand this question but I think it may be this formula:

2 pi 1 / 1 = 2 pi

But this would be divided by 4 since from this point to the next is ¼ of the overall circle. That is if I understand this correctly. 2 pi/ 4

confidence rating #$&*: 1

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Given Solution:

`aThe trip around the entire circle, which corresponds to an angular displacement of 2 pi radians, corresponds to a trip from the initial point to the point where the circle meets the positive y-axis (i.e., the point (0,1)), then from this point to the point where the circle meets the negative x-axis (i.e., the point (-1,0)), then from this point to the point where the circle meets the negative y-axis (i.e., the point (0,-1)), then from this point back to the point where the circle meets the positive x-axis (i.e., the point (1,0)).

Because of the symmetry of the circle, the arc corresponding to each of these displacements is the same. The arc from (1,0) to (0,1) is 1/4 of the 2 pi radian angular displacement around the entire circle, so its angular displacement is 2 pi/4 = pi/2 radians.

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Self-critique (if necessary):

I see where this makes sense and I also see that I forgot to simplify the problem to pi / 2

???I’m a little thrown off by the formula 2 pi r / r = 2 pi radians; wouldn’t this result in the answer always being 2 pie radians and making the radius measurement somewhat irrelevant???

At this time I’m assuming this concept will be further explained and used other ways in the future.

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Self-critique Rating: 3

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If you divide the arc distance by the radius, you get the number of radians.

An arc equal to the circumference has arc distance 2 pi r. Dividing this by the radius gives you the angle in radians, so 2 pi r / r = 2 pi radians.

You are correct that we'll be using this frequently.

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Question: `q009. We have just seen that the angular position of the (1,0) point is 0 and the angular position of the (0,1) point is pi/2. What are the angular positions of the (-1,0) and (0,-1) points?

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Your solution:

For the point (-1, 0) I believe this would be:

2 pi * ½

= 2 pi /2

= pi

And the (0, -1)

2 pi * ¾

= 6 pi / 4

= 3 pi / 2

confidence rating #$&*: 2

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Given Solution:

`aThese points are reached after successive angular displacements of pi/2. The (-1,0) point is reached from the pi/2 position by an additional angular displacement of pi/2, which puts it at angular position pi.

The (0,-1) point is reached after another angular displacement of pi/2, which puts it at pi + pi/2 = 2 pi/2 + pi/2 = 3 pi/2.

Note that still another angular displacement of pi/2 puts us back at the initial point, whose angular position is 0. This shows that the initial point has angular position 0, or angular position 3 pi/2 + pi/2 = 4 pi/2 = 2 pi, consistent with what we already know.

You should label your picture with these angular positions pi/2, pi, 3 pi/2 and 2 pi specified at the appropriate points.

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Self-critique (if necessary):

This is becoming easier to understand as I work through it.

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Self-critique Rating: 3

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That is my intent in sequencing these questions.

Given your demonstrated aptitude and work habits, ff it doesn't work this way in your case, I'll know the sequencing needs to be changed.

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Question: `q010. What is the angular displacement from the standard initial point of the point halfway along the arc of the circle from (1,0) to (0,1)? Note that you should begin with a sketch of the circle and of the arc specified here.

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Your solution:

This would be 1/ 8 of the circle there for making the angular displacement pi / 4

confidence rating #$&*: 2

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Given Solution:

`a(1,0) is the point at which the circle meets the positive x-axis and (0,1) is the point at which the circle meets the positive y-axis. The trip along the arc of the circle from (1,0) to (0,1) will move along the first-quadrant arc from angular position 0 to angular position pi/2. Halfway along this arc, the angular position will be 1/2 * pi/2 = pi/4.

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

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Question: `q011. What will be the angular positions of the arc points halfway between the (0,1) and (-1,0) points of the circle?

What will be the angular positions of the arc points halfway between the (-1,0) and (0,-1) points of the circle?

What will be the angular positions of the arc points halfway between the (0,-1) and (1,0) points of the circle?

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Your solution:

Halfway between (0,1) and (-1,0) would also be pi / 4 based on the calculations previously since this is 1/8 * 2 pi = 2 pi / 8 = pi / 4

Halfway between (-1, 0) and (0, -1) also is pi / 4

Halfway between the points (0,-1) and (1, 0) is once again pi /4

confidence rating #$&*: 2

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Given Solution:

`aHalfway between the (0,1) point, which corresponds to the the the position pi/2, and the (-1,0) point, which corresponds to angular position pi, will be the point lying at angular position pi/2 + pi/4 = 2 pi / 4 + pi / 4 = (2 pi + pi)/4 = 3 pi / 4.

Halfway between the (-1,0) point, which corresponds to the the position pi,and the (0,-1) point, which corresponds to angular position 3 pi / 2, will be the point lying at angular position pi + pi/4 = 4 pi / 4 + pi / 4 = (4 pi + pi)/4 = 5 pi / 4.

Halfway between the (0,-1) point, which corresponds to the the position 3 pi/2, and the (-1,0) point, which corresponds to angular position 2 pi, will be the point lying at angular position 3 pi/2 + pi/4 =62 pi / 4 + pi / 4 = (6 pi + pi)/4 = 7 pi / 4.

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Self-critique (if necessary):

I see where I misunderstood the question. I was making measurements between the 2 points as if the initial point had changed and not the particular point indicated from the original initial point.

But I do understand the reasoning behind the answers and the given concept.

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Self-critique Rating: 3

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Question: `q012. What is the angular position of the point lying 1/3 of the way along the arc of the circle between the points (1,0) and (0,1)?

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Your solution:

This would be 1 / 3 of ¼ of the whole circle and results in:

2 pi * 1/ 12 = 2 pi /12 = pi /6

confidence rating #$&*: 2

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Given Solution:

`aThe arc from (1,0) to (0,1) corresponds to an angular displacement of pi/2. One-third of the arc corresponds to an angular displacement of 1/3 * pi/2 = pi/6. The angular position of the specified point is therefore pi/6.

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Question: `q013. How long would it take to move around a complete circle at each of the following angular speeds:

• 1 radian / second

• 3 radians / second

• pi radians / second

• pi/4 radians / second

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Your solution:

I would think this would be found by taking the total radian calculation 2 pi r / r = 2 pi

And doing the following:

2 pi /1 = 2 pi seconds

2 pi / 3 = 2 pi /3 seconds

2 pi / pi = 2 seconds

2 pi / (pi/4) = 8 pi / pi = 8 seconds

confidence rating #$&*: 2

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Question: `q014. What is the angular position in radians of the point which lies 3/4 of the way along the arc of the circle which starts at (0, 1) and runs counterclockwise to the point (0, -1).

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Your solution:

Well this would be ¾ of ½ of a circle which the total radians is 2 pi so

2 pi* 3/8

6 pi /8

3 pi / 4

confidence rating #$&*: 2

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Question: `q015. What is the angular position in radians of the point which lies 2/3 of the way along the arc of the circle which starts at (0, 1) and runs counterclockwise to the point (-1, 0)?

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Your solution:

This is ¼ of a circle and we are traveling 2/3 of this distance making

1/6 * 2 pi

2 pi/6

Pi / 3

confidence rating #$&*: 2

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Self-critique Rating:

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Self-critique (if necessary):

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Self-critique rating:

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Self-critique (if necessary):

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Self-critique rating:

#*&!

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Very good.

Be sure to see my comments.

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