#$&* course MTH 164 9/25/14 around 7:25I had a question that I put under the self critique part of question 3. I got the answer right but the question was about your Note part of Your Answer. Question: `q001. Note that there are 10 activities in this assignment.
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Given Solution: `aWe see visually that the point a lies at an arc distance less than the radius of the circle. We also see that the point c lies at an arc distance that is clearly greater than the radius of the circle. The only possible candidate for a 1 radian angle, which must lie at an arc distance equal to one radius, is therefore point b. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `q002. If the first ant moves at a constant speed, moving through 1 radian every second, then approximately how long, to the nearest second, do you think it will take for the ant to move along the arc to the point where the circle meets the negative x-axis? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Points B, C, and D each seem to be 1 radian from each other and the initial point. If the ant is moving at 1 radian every second, then it should reach the negative x-axis just past 3 seconds. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating:3 ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If point B represents 1 arc radian and the and is moving at ½ arc radian per second, then after one second the ant should be at point A. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aAfter 1 second the angular position would be 1/2 radian, which would correspond to point a. Note that after 2 seconds the angular position would be 1 radian, corresponding to point b, and after three seconds the angular position would be 3 * 1/2 radian = 3/2 radian and the ant would be at position c. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I understand that the answer would be point A. However, on the note part, you put that after 3 seconds the ant would be at position C. But in question 2, you said that points B, C, and D would be at radians 1, 2, and 3 which is how I see it. ??? My question is that after 3 seconds, shouldn’t the ant be between points B and C somewhere close to the top of the circle. 3/2 radians would be 1 and ½ radians not 2. Sorry if I am looking at this wrong.???
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Given Solution: `aThe circumference of the circle is 2 pi r, where r is the radius of the circle. This is the distance traveled by the ant. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `q005. As we just saw the distance around the circle is its circumference 2 pi r, where r is the radius. Through how many radians would the ant travel from the initial point, where the circle meets the positive x-axis, if the motion was in the counterclockwise direction and ended at the original point after having completed one trip around the circle. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If Point D is about 3 radians, then the ant would travel about 6 radians if it went completely around the circle. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):I got the answer wrong because I thought you were talking about the circle from question 1. But after looking at your answer I understand how to do it and get the answer of 2 pi. ------------------------------------------------ Self-critique Rating: ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If the circle has a radius of 1 and centered on the origin, then it should meet the positive x-axis at (1,0), the positive y-axis at (0,1), the negative x-axis at (-1,0), and the negative y-axis at (0,-1). confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe unit circle has radius 1 and is centered at the origin, so the circle meets the positive x-axis 1 unit from the origin at (x, y) = (1,0). Similarly the circle meets the positive y-axis at the 'top' of the circle, 1 unit from the origin at (x, y) = (0,1); the circle meets the negative x-axis at (-1, 0); and the circle meets the negative y-axis at (0,-1). Figure 84 shows these points on the unit circle. Note that in this figure the small dots are located at increments of .1 unit in the x and y directions. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating:OK ********************************************* Question: `q007. Without looking at Figure 84, sketch a picture of the unit circle, complete with labeled points where the circle meets the x and y axes. Indicate the arc from the standard initial point, where the circle meets the positive x-axis, to the point where the circle meets the positive y axis. Describe your sketch. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The sketch is of an even t-graph of all four quadrants on the graph. The circle has a radius of 1 and meets the positive x-axis at (1,0), positive y-axis at (0,1), negative x-axis at (-1,0), and negative y-axis at (0,-1). The arc is from point (1,0) to (0,1) in the top right quadrant of the graph. confidence rating #$&*: OK ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating:3 ********************************************* 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: According to the class notes, there are 2 pi radians of arc length in a circle. So the distance from the positive x-axis to the positive y-axis should be 1/4th of the distance. So 2 pi radians / 4 would be pi/2 radians. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe trip around the entire circle, which corresponds to an angular displacement of 2 pi radians, corresponds to a trip from the initial point to the point where the circle meets the positive y-axis (i.e., the point (0,1)), then from this point to the point where the circle meets the negative x-axis (i.e., the point (-1,0)), then from this point to the point where the circle meets the negative y-axis (i.e., the point (0,-1)), then from this point back to the point where the circle meets the positive x-axis (i.e., the point (1,0)). Because of the symmetry of the circle, the arc corresponding to each of these displacements is the same. The arc from (1,0) to (0,1) is 1/4 of the 2 pi radian angular displacement around the entire circle, so its angular displacement is 2 pi/4 = pi/2 radians. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating:OK ********************************************* Question: `q009. We have just seen that the angular position of the (1,0) point is 0 and the angular position of the (0,1) point is pi/2. What are the angular positions of the (-1,0) and (0,-1) points? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If the entire circle is 2 pi radians and one quarter is pi/2, the at position (-1,0) should add pi to pi/2 = 2 pi/2 or just pi. Then adding another pi to get to point (0,-1) would give you 3 pi/2. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating:OK ********************************************* 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If the point from the initial point (1,0) to (0,1) is pi/2, then half of that should be pi/4. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a(1,0) is the point at which the circle meets the positive x-axis and (0,1) is the point at which the circle meets the positive y-axis. The trip along the arc of the circle from (1,0) to (0,1) will move along the first-quadrant arc from angular position 0 to angular position pi/2. Halfway along this arc, the angular position will be 1/2 * pi/2 = pi/4. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating:OK ********************************************* Question: `q011. What will be the angular positions of the arc points halfway between the (0,1) and (-1,0) points of the circle? What will be the angular positions of the arc points halfway between the (-1,0) and (0,-1) points of the circle? What will be the angular positions of the arc points halfway between the (0,-1) and (1,0) points of the circle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The angular position halfway between (0,1) and (-1,0) would be the adding pi/2 /2 to pi/2 or pi/4 to pi/2 which = pi /4 + 2 pi/4= 3 pi/4. The angular position halfway between (-1,0) and (0,-1) would be adding pi/4 to pi which would be pi/4 + 4 pi/4 = 5 pi/4. The angular position halfway between (0,-1) and (1,0) would be adding pi/4 to 3 pi/2 which would be pi/4 + 6 pi/4 = 7 pi/4. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating:OK ********************************************* 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)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If point (0,1) represents pi/2, then 1/3 of it should be pi/2 * 1/3 which would = pi/6. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe arc from (1,0) to (0,1) corresponds to an angular displacement of pi/2. One-third of the arc corresponds to an angular displacement of 1/3 * pi/2 = pi/6. The angular position of the specified point is therefore pi/6. ********************************************* Question: `q013. How long would it take to move around a complete circle at each of the following angular speeds: • 1 radian / second • 3 radians / second • pi radians / second • pi/4 radians / second YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If a complete circle is 2 pi radians then at 1 radian / second it should take 2 pi radians/1 radian/second = 2 pi seconds. At 3 radians / second then it should take 2 pi radians/3 radians/second = 2 pi /3 seconds. At pi radians / second then it should take 2 pi radians/pi radians/second = 2 seconds. And at pi/4 radians / second then it should take 2 pi radians/ (pi/4 radians)/second = 8 seconds. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* 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). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If you cut the circle into 8 sections so you can find 3/4 of the way between (0,1) and (0,-1) then each 8th of a section should be 2 pi/8. Then to get ¾ way between (0,1) and (0,-1), it would be 10 pi/8 which would be reduced to 5 pi/4. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q015. What is the angular position in radians of the point which lies 2/3 of the way along the arc of the circle which starts at (0, 1) and runs counterclockwise to the point (-1, 0)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If each quarter section of the arc is cut into 3 sections, then there would be 24 pi / 12 radians in a complete circle. Point (0,1) would be at 6 pi/12 or pi/2 and adding 2/3 from that point would be adding 4 pi/12 to 6 pi/12 = 10 pi/12 = 5 pi/6 radians. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating: Not sure if I am looking at the problem right.