course Mth 164 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): Need to become more specific with my anwsers. ------------------------------------------------ Self-critique rating #$&*:2 ********************************************* Question: `q002. If the first ant moves at a constant speed, moving through 1 radian every second, then approximately how long, to the nearest second, do you think it will take for the ant to move along the arc to the point where the circle meets the negative x-axis? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 3, because the ant has to move past three arcs in order to get to the negative side. confidence rating #$&*:: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q003. If the ant traveled at 1/2 radian per second, then after 1 second would its angular position be indicated by point a, point b, point c or point d in Figure 37? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This would put the ant in a correspondent postion to point A. confidence rating #$&*:: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q004. How far will the ant travel in the process of completing 1 trip around the circle, starting and ending at the initial point where the circle meets the positive x-axis. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The distance traversed would be the circumference of the circle 6.28(r). confidence rating #$&*: 4 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q005. As we just saw the distance around the circle is its circumference 2 pi r, where r is the radius. Through how many radians would the ant travel from the initial point, where the circle meets the positive x-axis, if the motion was in the counterclockwise direction and ended at the original point after having completed one trip around the circle. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It should come out to be 8 arcs. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q006. The unit circle is a circle of radius 1 centered at the origin. What are the coordinates of the points where the unit circle meets the positive x-axis, the positive y axis, the negative x-axis and the negative y axis? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Positive x (1,0) Negative x (-1,0) Positive y (0,1) Negative y (0,-1) confidence rating #$&*: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q007. Without looking at Figure 84, sketch a picture of the unit circle, complete with labeled points where the circle meets the x and y axes. Indicate the arc from the standard initial point, where the circle meets the x-axis, to the point where the circle meets the positive y axis. Describe your sketch. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: my sketch shows that the outer part of the circle hits (1,0), (0,1), (-1,0), (0,-1) on the axises of the graph. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q008. How many radians of angular displacement correspond to the arc displacement from the standard initial point, where the circle meets the x-axis, to the point where the circle meets the positive y axis? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the currcumference of the circle frome the initial point on the y-axis. confidence rating #$&*: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q009. We have just seen that the angular position of the (1,0) point is 0 and the angular position of the (0,1) point is pi/2. What are the angular positions of the (-1,0) and (0,-1) points? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: -pi and pi/-2 confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q010. What is the angular displacement from the standard initial point of the point halfway along the arc of the circle from (1,0) to (0,1)? Note that you should begin with a sketch of the circle and of the arc specified here. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the displacement is from the position 0 to pi/2 and halfway will be 1/2*pi/2=pi/4. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q011. What will be the angular positions of the arc points halfway between the (0,1) and (-1,0) points of the circle? What will be the angular positions of the arc points halfway between the (-1,0) and (0,-1) points of the circle? What will be the angular positions of the arc points halfway between the (0,-1) and (1,0) points of the circle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: pi/2 + pi/4 = 2 pi/4 + pi/4 = (2pi + pi)/4 = 3pi/4 pi + pi/4 = 4pi/4 +pi/4 = (4pi + pi)/4 = 5pi/4 3pi/2 + pi/4 =2pi/4 + pi/4 = (2pi + pi)/4 = 5pi/4 confidence rating #$&*: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q012. What is the angular position of the point lying 1/3 of the way along the arc of the circle between the points (1,0) and (0,1)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: pi/6 confidence rating #$&*: 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. "
course Mth 164 &&&& query modeling exercise
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Given Solution: ** At 3 rad/sec a complete trip around the reference circle takes 2 pi / 3 seconds, close to but not exactly 2 seconds. 2 pi / 3 seconds is the distance between the peaks on the graph of y vs. t. If the circle has radius 5 the max and min will be 5 units above and below the center of the circle, at 12 - 5 = 7 and 12 + 5 = 17. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& Given the values between which a cyclical quantity varies, how you determine where to position the circle that models the quantity, and how the determine the radius of the circle?
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15:22:13 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the center of the circle will be half way between the max and min values of the graph of the circle. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The center of the circle will be halfway between the max and min values, which can be found by averaging the two values (i.e., add and divide by 2). The diameter will be the difference between the max and min values and the radius will be half of the diameter. **
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15:22:16 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& What is the vertical coordinate of the center of the circle, and what is the angular velocity of the reference point, for the daylight model? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the min and max will be the hours in the day which the center is 12 and also the angular. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** If the period is 52 weeks then you have 2 pi / 52 cycles in a week or pi/26 cycles per week. If the period is in months then you have 2 pi / 12 cycles per month, or pi/6 cycles per month. The vertical coordinate of the center will be the day length midway between the min and max day lengths, which is 12 hours.**
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Given Solution: ** If the period is 52 weeks then you have 2 pi / 52 cycles in a week or pi/26 cycles per week. The vertical coordinate of the center will be the temperature midway between the min and max temperatures.**
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15:58:33 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):
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Given Solution: ** If you have a cycle in 10 hours then you have 2 pi rad in 10 hours, or 2 pi / 10 = pi/5 rad / hour. The vertical coordinate of the center will be the water level midway between the min and max water levels. **
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15:58:35 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& What is the vertical coordinate of the center of the circle, and what is the angular velocity of the reference point, for the ocean wave model? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: there are 5 waves/min and the frequency is 5per/min =10pi rad/min. halfway between the highest and lowest levels will lie the center. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The center will lie halfway between the highest and lowest levels. At 5 waves per minute the angular frequency would be 5 periods / minute * 2 pi rad / period = 10 pi rad / min. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& query ch. 5 # 78 15 in wheels at 3 rev/sec. Speed in in/s and mph: rpm?
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 3 rps in the wheels. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** If 15 inches is the diameter of the wheel then the radius is 15 inches. The angular velocity is 3 rev / sec * 2 pi rad / rev = 6 pi rad / sec. 905 pi rad / sec. If you approximate this you get around 280 in/sec. This is 280 in / sec * 1 ft / 12 in = 23 ft / sec approx. A mile is 5280 ft and an hour is 3600 sec so this is 23 ft/sec * 1 mile / 5280 ft * 3600 sec / 1 hr. = 16 miles / hr approx.. ** Note that 3 revolutions / second is 180 revolutions / minute, since there are 60 seconds in a minute.
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course Mth 164 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): Need to become more specific with my anwsers. ------------------------------------------------ Self-critique rating #$&*:2 ********************************************* Question: `q002. If the first ant moves at a constant speed, moving through 1 radian every second, then approximately how long, to the nearest second, do you think it will take for the ant to move along the arc to the point where the circle meets the negative x-axis? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 3, because the ant has to move past three arcs in order to get to the negative side. confidence rating #$&*:: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q003. If the ant traveled at 1/2 radian per second, then after 1 second would its angular position be indicated by point a, point b, point c or point d in Figure 37? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: This would put the ant in a correspondent postion to point A. confidence rating #$&*:: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q004. How far will the ant travel in the process of completing 1 trip around the circle, starting and ending at the initial point where the circle meets the positive x-axis. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The distance traversed would be the circumference of the circle 6.28(r). confidence rating #$&*: 4 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q005. As we just saw the distance around the circle is its circumference 2 pi r, where r is the radius. Through how many radians would the ant travel from the initial point, where the circle meets the positive x-axis, if the motion was in the counterclockwise direction and ended at the original point after having completed one trip around the circle. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It should come out to be 8 arcs. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q006. The unit circle is a circle of radius 1 centered at the origin. What are the coordinates of the points where the unit circle meets the positive x-axis, the positive y axis, the negative x-axis and the negative y axis? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Positive x (1,0) Negative x (-1,0) Positive y (0,1) Negative y (0,-1) confidence rating #$&*: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q007. Without looking at Figure 84, sketch a picture of the unit circle, complete with labeled points where the circle meets the x and y axes. Indicate the arc from the standard initial point, where the circle meets the x-axis, to the point where the circle meets the positive y axis. Describe your sketch. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: my sketch shows that the outer part of the circle hits (1,0), (0,1), (-1,0), (0,-1) on the axises of the graph. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q008. How many radians of angular displacement correspond to the arc displacement from the standard initial point, where the circle meets the x-axis, to the point where the circle meets the positive y axis? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the currcumference of the circle frome the initial point on the y-axis. confidence rating #$&*: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q009. We have just seen that the angular position of the (1,0) point is 0 and the angular position of the (0,1) point is pi/2. What are the angular positions of the (-1,0) and (0,-1) points? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: -pi and pi/-2 confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q010. What is the angular displacement from the standard initial point of the point halfway along the arc of the circle from (1,0) to (0,1)? Note that you should begin with a sketch of the circle and of the arc specified here. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the displacement is from the position 0 to pi/2 and halfway will be 1/2*pi/2=pi/4. confidence rating #$&*: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q011. What will be the angular positions of the arc points halfway between the (0,1) and (-1,0) points of the circle? What will be the angular positions of the arc points halfway between the (-1,0) and (0,-1) points of the circle? What will be the angular positions of the arc points halfway between the (0,-1) and (1,0) points of the circle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: pi/2 + pi/4 = 2 pi/4 + pi/4 = (2pi + pi)/4 = 3pi/4 pi + pi/4 = 4pi/4 +pi/4 = (4pi + pi)/4 = 5pi/4 3pi/2 + pi/4 =2pi/4 + pi/4 = (2pi + pi)/4 = 5pi/4 confidence rating #$&*: 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): ------------------------------------------------ Self-critique rating #$&*: ********************************************* Question: `q012. What is the angular position of the point lying 1/3 of the way along the arc of the circle between the points (1,0) and (0,1)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: pi/6 confidence rating #$&*: 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. "
course Mth 164 &&&& query modeling exercise
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Given Solution: ** At 3 rad/sec a complete trip around the reference circle takes 2 pi / 3 seconds, close to but not exactly 2 seconds. 2 pi / 3 seconds is the distance between the peaks on the graph of y vs. t. If the circle has radius 5 the max and min will be 5 units above and below the center of the circle, at 12 - 5 = 7 and 12 + 5 = 17. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& Given the values between which a cyclical quantity varies, how you determine where to position the circle that models the quantity, and how the determine the radius of the circle?
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15:22:13 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the center of the circle will be half way between the max and min values of the graph of the circle. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The center of the circle will be halfway between the max and min values, which can be found by averaging the two values (i.e., add and divide by 2). The diameter will be the difference between the max and min values and the radius will be half of the diameter. **
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15:22:16 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& What is the vertical coordinate of the center of the circle, and what is the angular velocity of the reference point, for the daylight model? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: the min and max will be the hours in the day which the center is 12 and also the angular. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** If the period is 52 weeks then you have 2 pi / 52 cycles in a week or pi/26 cycles per week. If the period is in months then you have 2 pi / 12 cycles per month, or pi/6 cycles per month. The vertical coordinate of the center will be the day length midway between the min and max day lengths, which is 12 hours.**
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Given Solution: ** If the period is 52 weeks then you have 2 pi / 52 cycles in a week or pi/26 cycles per week. The vertical coordinate of the center will be the temperature midway between the min and max temperatures.**
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15:58:33 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):
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Given Solution: ** If you have a cycle in 10 hours then you have 2 pi rad in 10 hours, or 2 pi / 10 = pi/5 rad / hour. The vertical coordinate of the center will be the water level midway between the min and max water levels. **
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15:58:35 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& What is the vertical coordinate of the center of the circle, and what is the angular velocity of the reference point, for the ocean wave model? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: there are 5 waves/min and the frequency is 5per/min =10pi rad/min. halfway between the highest and lowest levels will lie the center. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The center will lie halfway between the highest and lowest levels. At 5 waves per minute the angular frequency would be 5 periods / minute * 2 pi rad / period = 10 pi rad / min. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: &&&& query ch. 5 # 78 15 in wheels at 3 rev/sec. Speed in in/s and mph: rpm?
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 3 rps in the wheels. confidence rating #$&*: 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** If 15 inches is the diameter of the wheel then the radius is 15 inches. The angular velocity is 3 rev / sec * 2 pi rad / rev = 6 pi rad / sec. 905 pi rad / sec. If you approximate this you get around 280 in/sec. This is 280 in / sec * 1 ft / 12 in = 23 ft / sec approx. A mile is 5280 ft and an hour is 3600 sec so this is 23 ft/sec * 1 mile / 5280 ft * 3600 sec / 1 hr. = 16 miles / hr approx.. ** Note that 3 revolutions / second is 180 revolutions / minute, since there are 60 seconds in a minute.
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