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solve the system of linear equations.

Algebra and Trigonometry | 3rd Edition | ISBN: 9780470648032 | Authors: Cynthia Y. Young ISBN: 9780470648032 218

Solution for problem 58 Chapter 9

Algebra and Trigonometry | 3rd Edition

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Algebra and Trigonometry | 3rd Edition | ISBN: 9780470648032 | Authors: Cynthia Y. Young

Algebra and Trigonometry | 3rd Edition

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Problem 58

solve the system of linear equations.

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Visual Perception Midterm 2 Study Guide ____________________________________________________________________________ Research paper presentations Blue Highlighindicates key terms Yellow Highligh indicates researcher ­ Researchers to be familiar with: Nakayama, Haffenden/Schiff/Goodale, Witt et al., Woods et al., Proffitt et al., Durgin et al., Sinai et al., Potter, Gajewski et al., Kaufman & Rock ____________________________________________________________________________ Nakayama: ​ Gibson Appreciation ­ Gibson’s physics ­ Gibson interested in the physical world’s possible interactions with animal life ­ Surfaces ­ The surface is what touches the animal. Surfaces for Gibson are one of the most important features of the environment. ­ Gibson’s physics different from conventional physics in several ways: ­ Restricted to a scale comparable to the animal ­ Possessed a particularity not usually associated with physics, consisting of physical features of the environment relevant to animate life ­ Description of the physical world exclusively considered in terms of its potential functional relation to an animal’s existence ­ Gibson’s two main interest1)​How locomotion with respect to the physical environment is controlled, a​)​he consequences of locomotion for perception (optic flow*Reminder: ​ ptic flois the change of structured light in the image, e.g. on the retina, due to a relative motion between the eyeball or camera and the scene ­ Main takeaway of Gibson’s physics: ​ibson’s physics not about the physical world by itself, but only in its relation to potential physical actions of animals ­ Locomotion ​= one class of actions common to all animals Psychology of Information Pickup Gibson concerned with surfaces in particular reference to the actions of animals ­ How do our eyes interact with the environment ­ It was once thought that vision was static ­ Gibson suggests that it is more dynamic – “even a slight movement of a photographer’s camera will shift the image and ruins the picture” comparison ­ Light is not just a stimulus, but also a carrier of information ­ Relative motion and the control of locomoti​ptic flow provides sense of direction during movements Final Takeaways ­ Despite not participating in research and his lack of interest in psychophysiology, Gibson was able to provide a framework for the field of visual perception ­ Talked about the importance of motion­based vision; the importance of the observer being able to actively interact with the world – actively engaging with the environment and not just passively observing ­ Gibson was seen as radical because at the time, vision was thought to be static, and he said instead that vision​ynamic and ​ctive​rocess ­ Biggest pointinformation picku​how do humans take in visual cues) ­ Said our eyes constantly pick up information about gradient, texture, motion, etc. because our eyes have properties that can observe these ­ Legacy: visual experiments that mimic the real world changed the static notion of vision into a dynamic and active one ____________________________________________________________ Perception and Action ​­ A Quick Review ­ The “what” stream is the v ​entral stream ­ The “where” stream is the d ​ orsal stream ­ Example: Patient DF had damage in her ventral stream, and therefore had trouble using vision to recognize objects, but she could use her vision to reach out for something (the “where” factor) ____________________________________________________________ Haffenden, Schiff, & Goodale ​(2001) ­ Studied the dissociation between perception and action in the Ebbinghaus illusion ­ Background ­ 1) Recently proposed distinction: vision for perception and vision for action ­ Visually guided movements should be largely immune to the perceptually compelling changes in size produced by pictorial illusions ­ 2) Previous tests: small effects of illusion on action using Ebbinghaus illusion ­ There is a single ​(not double) presentation of size for perception and action ­ 3) More recent findings: confounding variables ­ 2­D pictorial elements influence vision for action: give rise to an effect on grasp ­ 4) Gap between target and illusory­making elements is equidistant across different conditions ­ In contrast, grasp scaling appeared to be affected primarily by the physical proximity of the 2­D illusory elements to the target, regardless of the size of the surrounding elements ­ Experiment ­ materials and method ­ 18 subjects ­ Viewing period controlled by goggles with liquid­crystal shutter lenses that remain opaque until they receive a switch signal from the experimenter (this change took 2.5 ms) ­ Markers of three­camera Optotrak system were fastened to index fingers, thumbs, and wrists to record finger and hand position ­Stimuli­target disks ­ Ebbinghaus illusion ­Traditional small: ​ traditional small circle surrounded by 11 smaller circles ­ Adjusted small:​ adjusted small circle surrounded by 16 smaller circles ­ Traditional large:​traditional large circle surrounded by 5 large circles ­Procedure ­Each subject performed two tasks: a manual estimation task and a grasping task Manual estimation task ­Lens clear at first to allow participants to view the display ­After seeing the display, the subjects should move their hands to their body and manually estimate the size of the disk by separating their thumb and index finger until they felt the gap accurately match the width of the target disk they had just seen (hold position for 2.5 ms) Grasping task ­Similar procedure: After the display was clear to view, the subjects should immediately reach out their hands to grasp the target disk with a neutral movement Effects of illusory displays on estimations/grasp scaling ­ From smaller to larger surrounding circles in the manual estimation task, participants consistently guessed smaller ­ Similar trend in grasp task, except between the adjusted small and traditional large illusions ­ Primary finding of the study:​rasp was influenced by the​istance​between the center circle and the surrounding circles, NOT ​izeof the surrounding circles Effects of target disk size on estimations/grasp ­ Significant interaction between task and target disk size ­ The size of the grasp AND the size of the estimation increased as the target disk size increased Takeaways from this study ­ It has been suggested that small changes in grasp reveal that there is a single representation of size for b​terception​(manual estimation) and​ction(grasp) → This experiment shows that this is not true ­ There is a distinction between how we perceive size as it relates to perception and action → ​istance​ is a key variable ____________________________________________________________ Witt et al. Studied the effects of effort on the perception of distance ­ Perception of spatial layout is a function of optical information, and a function of the perceiver’s potential to act on the environment ­ Perception of distance is a function of distal extent + the action that the perceiver tends to perform + effort associated with this action ­ Rationale of the current experiment ­ Apparent egocentric distance increased when participants wore a heavy backpack ­ Changing the relationship between walking effort and concurrent optic flow produced a visuomotor effect ­ A recalibration between walking energy and anticipated optic flow changes the anticipated effort required to walk a given extent ­ Increasing anticipated walking effort led to increases in estimated distance ­​Prediction of the current study: P ​ erceived distance should increase after walking on the treadmill (increase in effort) if participants were intending to walk but not if they were intending to throw Experiment 1 ­ Will anticipated effort for throwing influence distance perception ­ Participants assigned to heavy ball or light ball condition ­ Participants threw either ball to 4 targets ranging in distance from the participant (3­11m) ­ After throwing the ball 3 times, participants were asked to estimate distance to the target ­ Results: ​Participants in the heavy ball condition perceived the targets as being farther away than did those who threw the lighter ball Experiment 2 ­ Throwing effort and perceived distance: perceptual matching ­ Design: ​participants were either assigned to the heavy ball or the light ball condition. They were also assigned to one of two conditions: “in” or “out.” ­ “In” condition: ​poles were initially placed directly behind the target and moved towards the double marks ­ “Out”​ poles were initially placed on the doubled marks and moved towards the target ­ While holding the ball, participants gave instructions to the experimenters about whether to move towards or away from each other until they thought that the distance between the two poled matched the distance from the observer (them) from the target (egocentric distance) ­ Results: ­ The main effect for ball was significant. Participants who threw the heavy ball positioned the poles to be farther apart than participants who threw the light­ball. ­ The main effect for pole condition was also significant. Participants in the “in” condition positioned the poles to be closer than participants in the “out” condition. ­ The only interaction that was significant was the interaction between pole condition and distance ­ Results show that effort for throwing influences apparent egocentric distance judgments Experiment 3 ­ Throwing effort and perceived distance: blind walking ­ Design: ​assigned to either heavy ball or light ball condition. Each participant blind walked to the 6 target distances ­ Procedure: ​ Participants were instructed to throw to each target until they hit it. They were then turned 180 degrees to face a new direction so they would respond to distance and NOT location of the target with blindfold on. They then attempted to walk the perceived target distance ­ Results: ​ The effect of effort for throwing does not extend to blind walking. The only time that perception is influenced is by throwing and not by walking. Experiment 4 ­ Effort for throwing and intention ­ Design: ​ Experiment 4​ looked at how effort of throwing and intentions affected participants’ perceived distance. Participants were assigned to either the “intend­to­walk” group or the “intend­to­throw” group ­ Those in the “intend­to­walk” group anticipated that after estimating the distance of a target, they were to walk to it ­ Those in the “intend­to­throw” group anticipated that after estimating the distance of a target, they were to blind throw a heavy ball to it ­ Procedure: ​ For the 4 test targets, after estimating the distance of the target, half the participants were given the ball and asked to close their eyes and throw to the target. The other half of the participants was asked to put on a blindfold and to walk to the target. ­ Results: ​ The experiment found that participants in the “intend­to­throw” group consistently overshot the estimated distance ­ These results suggest that the effect of effort for throwing on conscious distance perception is functionally specific, and depends on the action the perceiver intends to make Experiment 5 ­ effort for walking and intention ­ Design: ​ This experiment looked at the effort of walking and intentions affected participants’ perceived distance. Participants were once again assigned to either the “intend­to­walk” group or the “intend­to­throw” group ­ Those in the “intend­to­throw” group anticipated that after estimating the distance of a target, they were to throw a heavy ball to it those in the “intend­to­walk” group anticipated that after estimating the distance of a target, they were to walk to it. ­ In this experiment, experimenters manipulated anticipated walking effort ­ participants were placed on a treadmill, received zero optic flow ­ Procedure:​ In the pre­test round, participants estimated the distance of three targets 6, 8, and 10 m ­ Next, participants completed their walk or throw task. In the pre­test round, participants in the “intend­to­walk” group walked on the treadmill without optic flow for their estimated distance. ­ In the test round, target was placed at 8 m away. Participants removed the blindfold, estimated the distance to the target, and then either blind walked or threw to the target. ­ Results: ​ Relative to the pretests, participants in the blind walk condition overestimated the target in the test condition while participants in the throw condition underestimated the target in the test condition. ­ These results suggest that perception is influenced by the effort associated with an intended action, and thus is functionally specific. ­ Conclusion: ​ Perceived distance is a function of: 1) actual distance as specified by optical variables, 2) what we are intending to do, and 3) the effort associated with this intended action ____________________________________________________________ Woods et al. Overview ­ It has been suggested that objects appear farther away when more effort is required to act upon them (e.g. throwing a ball) ­ Authors of this study found nothing to suggest that this is true ­ However, effort in verbal reports was affected when participants were asked to take into account cognitive (nonvisual) factors ­ Authors believe that this is because participants were encouraged to take into account non­perceptual connotations of distance while not changing distance at all Experiment 1 ­ 24 participants were alternately assigned to either backpack or no­backpack condition ­ Each participant made 24 distance estimates. They divided these estimates into two blocks of six practice trials, followed by two blocks of six test trials. Six stimulus distance were presented in random order in each block ­ The target was a 0.23 m tall cone ­ The experiment took place on a flat, grassy field (120 m / 100 m). Golf tees marking target locations for experimenters were not visible to participants ­ Participants held a 12­inch ruler as a scale reference throughout testing. Participants stood at a central location for the duration of the study; targets were presented in one of six possible directions ­ They then turned to view the target and verbally reported the distance from themselves to the target ­ Predicted that participants wearing a weighted backpack would verbally report distances as being significantly greater than those not wearing a backpack ­ Results: ​ No effect of backpack Experiment 2 ­ The experiment was performed indoors in a space that contained some irregularly spaced floor markings. Targets to throw the ball to were presented along 6 radii, which were divided across two starting locations. The experimenter would then roll the ball back to the participant for the next throw. ­ Participants assigned to heavy ball or light ball condition and gave 12 estimates ­ Results: ­ Subjects expended appropriate amount of effort in throwing the ball ­ No effect of ball weight on verbal distance estimates Experiment 3 ­ Purposes were to evaluate whether intention to act was the essential missing component in preventing replication in Experiment 2 and what effect the inclusion of such a component had on subject responses (essentially the difference between Experiments 2 and 3) ­ 24 Participants were assigned in alternating order to either a heavy­ball or light­ball condition (same design as in Experiment 2) ­ Procedure differed only in that after verbally estimating a particular target distance, participants lowered a blindfold and immediately attempted to throw to that target. ­ Procedure differed only in that after verbally estimating a particular target distance, participants lowered a blindfold and immediately attempted to throw to that target. ­ Results: ​ Despite the inclusion of an intention to act component, there was once again no effect of ball weight on verbal estimates. ­ Inclusion of an intention component significantly increased verbal estimates of distance Experiment 4 ­ No effect of the effort manipulation on verbal estimates ­ Participants who were tested indoors gave significantly larger verbal estimates of distance than participants who were tested outdoors ­ However, there was again no effect of ball weight Experiment 5 ­ Performed a 2 (ball weight) * 2 (sex) * 3 (instruction group) * 2 (test block) * 4 (distance) repeated­measures multivariate analysis of variance ­ Neither the objective­distance nor apparent­distance instruction groups demonstrated any between subjects effects ­ The nonvisual­factors group demonstrated a between­subjects effect of ball weight ­ Heavy­ball participants gave significantly larger verbal distance estimates than light­ball participants ­ Blind­throwing performance ­ Repeated­measures of multivariate analysis of variance involving the 4­m and 6­m blind­throwing trials showed no ball weight effect in any of the instruction groups ­ Perceived distance was not affected by the effort manipulation ­ Implications: ­ Nonvisual­factors instruction did yield significantly different verbal responses than did the other two instruction conditions and was the only condition associated with effort­related effects ____________________________________________________________ Proffitt et al. Introduction: Geographical Slant ­ Geographical slant ​is the inclination of surfaces relative to the environmentally specific horizontal ­ 3 different ways surface slant can be defined: ­ Relative slant: specifies the orientation of one surface with respect to the reference frame provided by another ­ Optical slant: specified in relation to the line of sight from the point of observation to the surface in question ­ Geographical slant:​ independent of viewpoint; its magnitude is specified in relation to the horizontal ­ Pitch​ is defined as the surface’s rotation away from horizontal around the x­axis slant is equivalent to the pitch angle that the surface makes with respect to the ground ­ Slant is equivalent to the pitch angle Perceiving geographical slant ­ Most studies done investigating optical slant ­ studies found similar errors in estimating pitch ­ Optical slant perception studies focus on how selected information affects pitch perception vs. everyday geographical slant perception ­ A common finding in all of these studies has been that surfaces appear to lie closer to the parallel plane than the perspective projection indicates – thus, subjects have systematically underestimated the pitch in relation to the picture plane ­ Underestimation from the picture plane is equivalent to overestimation from the ground plane ­ Generally, it has been thought that these underestimations are due to a tendency of judgments to conform to the pitch of the projection surface or reduction screen ­ The source of error seems to lie in perceptual processes as opposed to the availability of reliable information ­ Similar to those found for the perception of geographical slant: Perceived pitch is invariably overestimated in relation to the horizontal ground plane (underestimated relative to the vertical picture plane) ­ Although people overestimate geographical slant, they are not especially accident prone when walking up and down hills – people are able to accurately judge the maximum inclination that they can walk up, and people are also good at matching the pitch of a distal surface with that of a haptically (haptic = by touch) perceived surface Overview of experiments ­ Five conducted ­ 1st ­ geographical pitch judgments from people at bottom of hills using three different types of measuring perception ­ 2​d – replication of the first, except hill viewed from the top ­ 3​ and 4​t – replicated first and second but virtually ­ 5​t – assess perception before and after fatigue Experiment 1 ­ real hills viewed from the base ­ Methods:​ 300 participants. Nine hills at UVA ­ Needed to be near foot traffic ­ Top of the hill had to be over the horizon ­ Even surface ­ Apparatus:​ Three tests: ­ Verbal – said how big they perceived the pitch to be in degrees ­ Visual – used a disk with an adjustable angle to represent the cross­section of the inclination of the hill; subjects adjusted the disk to what they thought best represented the pitch of the hill while holding it approximately perpendicular to their line of sight ­ Haptic (by touch) ­ Procedure: ​ Part 1 ­ Viewed hills from the front while standing at the base ­ Instructed to look directly ahead and not to look at the sides ­ Judge angle of inclination of the hill with respect to the horizontal using the three tests in randomized order Part 2 ­ Test subjects for internal consistency ­ Asked to adjust either the tilt board or disk to a set of verbally given angles ­​Results ­ Subjects’ verbal and visual perceptions reflect large overestimations of the actual inclines; haptic perceptions were more accurate ­ Haptic reports were not significantly different from the actual inclines of the hills ­ Visual and verbal reports were overestimations of the actual pitch angle of the hill ­ In given angle tests: haptic measurements underestimated, and visual measurements more accurate ­ Overall, subjects displayed an internal consistency for the verbal, visual, and haptic measures, in that they represented an angle in the same way regardless of whether it was made in the context of an observed hill or of a spoken number Experiment 2: r ​eal hills viewed from the top ­ Hypothesis: ​ The perception of geographical slant relates distal inclines to our behavior potential ­ Same method and procedure as first experiment, except subjects were now looking down ­ Results: ­ Same as Experiment 1 as the verbal and visual judgments reflected large overestimations of the inclines and the haptic reports were far more accurate ­ Verbally given angles: visual responses close to accurate and haptic responses were underestimated ­ Subjects were consistent in their visual and their haptic judgments, meaning that the judgments that they made in response to the verbally given angles were nearly equivalent to the judgments that they made for the hills, given their verbal reports on the inclination of the hills. Experiment 3 ­ h ​ills viewed from the base in virtual reality ­ Purpose: obtain normative geographical perception data over a wider range of hills than can be naturally observed outside ­ Methods:​ Stimuli: 12 simulated hills ranging from 5 to 60 degrees displayed in full color with a checkerboard road running up the middle of each. A “post” was located 10 meters from the base of each hill. The subjects’ viewpoint was at the base of each of these hills. ­ Apparatus:​ 2 VGX computers that projected images into the head mounted devices (HMD) creating the virtual reality (VR) and a tilt board that corresponded in reality with the 10 meter post seen in VR ­ Design: ​ participants asked to gauge according to two measures: Verbal and Haptic ­ Each subject saw all of the hills in a random order and, similar to Experiment 1, were told to verbally state to what degree they perceived the hill pitch/slant to be within the 5 to 60 degree range (verbal measure) ­ Additionally, each participant was asked to tilt the tilt board in front of them (they couldn’t see the board due to the VR headset they were wearing), to the degree of pitch they perceived the displayed hill to be (haptic measure) ­ Results: ­ Overall, participants tended to overestimate the hill’s pitch when using a verbal measure and slightly underestimated the hill’s pitch when using the haptic measure, though the haptic measure tended to be more accurate for each participant ­ The verbal and haptic measures differed slightly, with the haptic measure being closer to the actual degree of the hill’s pitch displayed ­ However, the lines are almost parallel which speaks to the internal consistency between the verbal and haptic measures of each participant Experiment 4 ­ Hills viewed from the top in virtual reality Purposes: ­ The primary purpose of this experiment was to determine whether viewpoint would interact with pitch judgments as it had across Experiments 1 and 2 ­ Obtain pitch angle judgments from the tops of hills for inclinations not readily available in the real world ­ Assess the correspondence between the judgments for real hills and those presented in VR ­ The ​primary question ​that the researchers were attempting to answer in this experiment was:​ Would the steepest hills look steeper from the top than from the bottom ​ They wanted to know if the change in perspective would affect the participants’ perception of hill pitch/slant. ­ Design identical to Experiment 3’s, except the perspective of the participant was now atop the hill as if they were standing on it ­ Results: ​Verbal judgments less accurate than haptic judgments ­ The verbal judgement far overestimates the haptic judgment (that’s the judgment derived from the physical tilting of the tilting of the tiltboard to the perceived pitch of the hill displayed in their headsets). This trend holds mostly constant from Experiment 3 even when the participants’ perspective was shifted to the top of the hill. Experiment 5 ­ Effect of fatigue on perceiving geographical slant ­ 5° and 31° hills ­ The subjects were told that they would be required to go on a run of their choice and would be answering questions both at the beginning and at the end of their runs. ­ They would be given the starting and finishing points for their runs, which were the 5° hill and the 31° hill, respectively, for half of the students ­ The other half started at 31° and finished at 5° ­ Instructed to go on run of their choice as long as they go from start to finish, and end very fatigued ­ Distracter questions were asked as well as questions about inclination at first hill and second hill ­ Results: ­ 5° Hill ­ Verbal and visual estimates of inclination greater after run ­ Haptic judgments had no significant change prior to and following run ­ 31° Hill ­ Similar results ­ Experiment 1 ­ Similar results prior to run Takeaway Points ­ Geographical pitch grossly overestimated in visual and verbal measures ­ Important function is to inform planning/modulation of gait so as to expend energy at a desired rate ­ Haptic measures of pitch show little evidence of slant overestimation – people are able to walk up hills without stumbling. Due to: ­ Actions are mediated by visual awareness following a transformation that compensates for overestimations ­ Presence of separate visual pathways ­ Hills perceived as steeper when viewed from top and/or when fatigued ____________________________________________________________ Durgin et al. Geographical slant, two­systems theory, separate accurate motor representation of hills ­ Palm Board Method ­ Used in the two­systems theory approach ­ Durgin:​ adjusting geographic slant of palm boards is NOT visually guided ­ Palm boards depend on wrist flexion (creating bias) ­ Two­systems theory approach – invalid Hypothesis: ​ Palm board accuracy is related to the need for motion action to be accurately guided Experiment 1: Palm board bias ­ Not within reachable distance ­ ​0 degree incline ­ Palm board condition usually set 10 degrees lower than free hand condition ­ Verbal responses recorded ­ Supported idea that palm board is biased Experiment 2: Replication with an outdoor hill ­ Same procedure as Experiment 1 ­ Freehand condition less accurate than in Experiment 1, but still more accurate than palm board condition Experiment 3: Palm board low slant estimates ­ Palm board greatly underestimated the slope of the stimuli presented ­ Freehand condition – much more accurate ­ Condition done with the stimuli within a reachable distance Experiment 4: Proprioception of wrist flexion is similarly biased ­ Proprioception:​ being aware of the position of your body parts ­ Participants’ posture was manipulated; some were tested with classic posture and the rest were tested with palm board just above their navel ­ Materials: ​3 campus paths ­ Procedure: ​ Randomized order of 3 hill presentation; participants turned around and closed their eyes; fixed palm board to match their memory ­ Results: ​ Small changes in palm board heights produce large changes in palm board estimates ­ Discussion ­ Palm boards are biased and insensitive measures ­ Instead of the two­systems theory, the experiments and research suggest that there exists only one exaggerated representation of geographical inclination Takeaway point: ​ Durgin argues that the process of perceiving geographical slant perception is not separated within the brain ____________________________________________________________ Sinai et al. Main research question:​ does the brain select the most resource and time­efficient way to represent the three­dimensional location of objects ­ More specifically, does the brain use a quasi­two­dimensional coordinate system (X, Y) with respect to the ground surface (s), rather than a three­dimensional coordinate system (x, y, z),in order to simplify and speed up computations Experiment 1 ­ Theory: ​ When the ground surface is disrupted, the visual system can’t establish a reliable reference frame and fails to obtain correct absolute distance ­ If the texture of the ground surface is changed or there is a gap in the ground surface, the individual won’t be able to determine absolute distance ­ Methods: ​ Manipulation: target on other side of gap in the ground from a naive observer ­ Procedure: ​ Blindfolded, turned 90 degrees, asked to walk distance equivalent to remembered absolute distance of the target ­ Asked to judge the absolute distance of a target, and then to perceptually set the distance of a matching target to be at an equal distance Experiment 2 ­ Theory: ​When the ground surface is disrupted, the visual system can’t establish a reliable reference frame and fails to obtain correct absolute distance ­ Methods: ​ Manipulation: target on other side of a larger and deeper gap in the ground from a naïve observer ­ Procedure: ​ Blindfolded, turned 90 degrees, asked to walk distance equivalent to remembered absolute distance of the target ­ Asked to judge the absolute distance of a target, and then to perceptually set the distance of a matching target to be at an equal distance. Experiment 3 ­ Theory: D​ istance/texture model: Z = (H x G)/3 ­ Z is perceived absolute distance ­ H is the observer’s eye height relative to the ground surface ­ G is the local texture gradient on the ground at the target’s location ­ Methods: ​ Observers stood on an elevated ground surface (2.0 m) and estimated horizontal distance of a target placed on a lower ground surface ­ 8 participants were tested using the blindfolded test and 5 were tested using the perceptual distance matching test Experiment 4 ­ Distance/texture model: Z = (H x G)/3 ­ H is the observer’s eye height relative to the ground surface ­ G is the local texture gradient on the ground at the target’s location ­ Methods: ​ Two distinct texture regions: grass, concrete. Observer stood on concrete, previewed target on grass, then did the blindfolded walking task. Takeaways ­ The internal representation of space, which is based on the common ground surface, is used for both space perception and visually directed performance ­ Texture gradient of size + eye height influences judgment of absolute distance. Texture gradient on the ground acts as a depth cue for visual system to establish a reference frame. ____________________________________________________________ Potter Purpose of the paper: ​ Provide a preliminary analysis of the mind’s ability to scan the environment through brief glimpses and attach meaning to the images observed ­ To determine if people can detect an EXPECTED scene even when presented briefly ­ To determine the form of information people require in anticipation of the scene in order to codifier the resulting information accurately ­ The pace at which the human eye captures images for the brain to interpret is brief but the associated meaning from observations made do not always match or become translated appropriately due to the limited time of observation ­ Glances are too brief to ensure memory for what is observed Visual search and meaning ­ Average eye fixation lasts 1/3 of a second. Novel scenes or images presented at this high rate do not look instantly familiar ­ Each glance is too brief to assure memory for what is observed; however, ​ Potter believes that the briefest glance can be sufficient to trigger memory if the observer is aware of what to expect ­ In the study, observers are exposed to a succession of rapid glances using photographs of various scenes ­ In some instances, they had foreknowledge of the observations to be made and in others they did not Hypothesis 1: ​ When an observer knew precisely the features of an anticipated scene, he/she makes a direct visual match Hypothesis 2: ​ When an observer only has general information regarding the scene, he/she needs to recognize and categorize the multiple information received in order to make a positive match Experiment ­ One practice and 8 test sequences of 16 color pictures were shown on a projector to two groups of participants ­ A succession of rapid glances around the environment was stimulated by presenting observers with a sequence of various scenes and objects ­ Observer was instructed to look for a particular picture­ target picture ­ Group 1: Observer was shown the target picture before viewing each sequence ­ Group 2: Observer was only given a name for the target picture. The names were brief descriptions of the main objects/events in the target photo. ­ When the observer saw the picture, he/or she responded by pressing a lever that stopped the projector Results ­ Exposure time of only 125 msec to the target stimuli was representative of 70% of target detection ­ Due to the small time of exposure, one may assume that the participant scanned only for pictures ­ Prior research has shown that pictures are very effective memory cues ­ However, although identification of pictures takes only a meager 125 msecs, retention of pictorial memory took upwards of 300 msecs ­ Awareness of the meaning of a target stimuli is as important knowing the exact visual appearance of an object Conclusion ­ Perceiving the emergence of a target did not help in knowing its meaning, which suggests people process what they see rapidly to an abstract level of meaning before choosing to either accept or reject the interpreted meaning ­ Without extended time to think on them, unselected scenes are forgotten ­ Therefore, the ⅓ second length of an average glance is a delicate time in which an observer balances the need to scan his/her environment for desirable objects and the need to retain the useful information from the unwanted information ­ The hypotheses were accurately proven, thereby indicating the processing rate of the mind from observations through the eye to the provocation of an idea is rapid ____________________________________________________________ Gajewski et al. Introduction ­ Speed of Extraction ­ The s​peed of extraction ​​s the amount of time it takes for the visual system to extract (comprehend) information from a scene ­ To extract visual information from the environment, the eyes are directed from one location to another at a rate of 3x per second ­ Although gathering visual information about multiple objects requires multiple gaze shifts, a person can determine the gist of a scene on the basis of information within the time frame of a single fixation ­ Absolute distance​ (also called​gocentric distance)​, the distance between an observer and an object, is one element of a scene for which the speed of extraction has not yet been determined ­ Little is known about the time it takes to extract distance information from real­world environments ­ Information­processing speed can be affected by old age and brain injury, but even in individuals without brain damage, highly dynamic tasks such as driving can create conditions that limit one’s ability to fully extract distance information from the environment Experiment 1 ­ Experimenters compared blind­walking performance based on a viewing duration that was too short for quick eye movement (113 ms) with blind­walking performance based on an extended viewing time (5 s) ­ Blind­walking was used to demonstrate the visual system’s ability to extract spatial information ­ Participants viewed objects through a liquid crystal shutter window, which can transition between a translucent state and a transparent state very quickly ­ Timing accuracy of the shutter window was controlled by the experimenters ­ A masking stimulus was used to ensure performance was based on the glimpse through the shutter window during the experiment, rather than memory of the environment ­ Procedure: ​ The masking stimulus was projected onto a screen to the observer’s left, and this image reflected onto a beam splitter (which is like a two­way mirror) ­ The experiment environment was a slightly darkened room in a lab space that was 7.4 m away from participant’s viewing position ­ An orange cone was the target object presented at 11 distances ranging from 3 m to 5 m ­ The two viewing conditions, brief (113 ms) and extended (5 s) were administered through 11 trials each ­ When the shutter window opened, the participant viewed the target object, then the shutter closed after either 113 ms or 5 s (depending on the condition) ­ The participants then walked through the darkened room to the distance at which they thought the cone was placed ­ Results: ​ Performance generally good; ­ Differences between viewing duration depended on block order ­ When the 113 ms block was first, there was a greater bias toward underestimating the distance with brief viewing than with extended viewing ­ The bias largely disappeared with extended viewing, thus reflecting differences in what can be extracted during the two viewing times ­ In contrast, performance during the two viewing conditions was indistinguishable when the 5 s block was first ­ These results suggest that the information needed to support performance can be extracted in a single brief glance when longer viewing duration glances are provided beforehand Experiment 2 ­ Purpose: ​ To determine the shortest viewing duration that would support a response sensitive to the distribution of target distances employed ­ Experiment 2 examined performance at an ultra­brief viewing duration (9ms) and moderately brief viewing duration (69ms) ­ Method: ​ Two groups of 12 naive observers ­ First group performed 9­ms block of trials first and second group performed 69­ms block first ­ A Styrofoam ball was placed on the floor as a target at nine distances in the 3 to 5m range ­ Results: ​ Participants exhibited nearly perfect response sensitivity ­ The outcomes did not depend on viewing duration or block order ­ The results confirmed that target detection in 9­ms viewing condition was highly reliable Experiment 3 ­ Purpose: ​ Removed sources of distance information from the viewing situation ­ Removed angular declination that has shown to be potent cue within the distance range ­ Method: ​ One group of 12 naive observers ­ Target was presented at eye level with viewing durations of 9ms, 65 ms, and 187 ms at nine distances ­ Results: ​ Accuracy was generally high but performance did improve across the range of viewing duration ­ Although there was a modest drop in response sensitivity and a bias toward overestimation in 9ms, observers clearly extracted useful information about distance in this duration even in this reduced­cue setting Experiment 4 ­ Purpose: ​ To examine performance with eye level objects: each with angular size rendered non­informative ­ Done by systematically varying physical size of the target with distance so target always subtended to the same visual angle ­ Method: ​ 2 groups of 12 naive observers. Same as 2 and 3, but angular size remained constant. 2 blocks of 15 trials. ­ Floor level blocks vs eye level blocks ­ Target = five styrofoam balls ­ Results: ​ Clear performance difference between floor level and eye level ­ Accuracy did not depend on block order = bias for response sensitivity ­ Greater bias toward underestimation of floor target than with eye target ­ Effect of duration for floor and eye viewing conditions ­ No effect on bias or response sensitivity for the floor level targets and eye level targets ­ Useful information about distance can be extracted from glimpses as brief as 9 ms when relative angular size or angular declination is informative = one of the more extraordinary outcomes ­ Discussion: ​ What duration of viewing is required for observers to determine the distances to objects in real­world environments Accurate response depended upon: ­ Visual cues ­ Stored information about the environment ­ Floor target ­ more than a brief glimpse ­ Participants’ sensitivity to change in physical size Implications for theories of scene perception and space perception ­ Domain = primarily on the directional and conceptual locations of objects as opposed to their absolute distance ­ Shown that the time needed to extract distance information varies widely depending on the kind of information available ­ Absolute distance perception = dynamic even when static and stable head position ____________________________________________________________________________ Kaufman & Rock The Moon Illusion ­ Two theories exist to explain why the moon at the horizon is larger than the moon at its zenith: ­ Apparent­Distance Theory: ​horizon moon looks larger due to it seeming farther away ­ Angle­of­Regard Theory: zenith moon looks smaller due to the elevation of the viewers’ eyes/head Angle­of­Regard Theory – Boring ­ Postural positions affect our judgment of the size of objects (i.e. neck craned or viewed straight on) ­ Conducted a series of experiments which allowed viewers to see both horizon and zenith moon “straight on” and with “eyes raised” ­ Subjects lifted neck to view zenith moon straight on and laid supine to view horizon moon with “eyes raised” (manipulated posture to produce illusion) ­ Found that “straight on” view correlated with illusion of larger size Challenges to Angle­of­Regard Theory ­ Kaufman & Rock (1962) challenged Boring’s method of producing the illusion ­ Proposed that even with the manipulated postures, it was impossible to compare the moon to disk sizes because the objects were not “commensurable,” or not measurable by the same standard due to the moon’s more or less “indeterminate size” over a seemingly “infinite” distance ­ Hypothesized that size would influence the perception of what moon was nearer or farther. When the zenith moon was larger than the horizon moon, participants said it was nearer and vice versa ­ Came to the conclusion that it had to do with the size as well as the terrain Apparent Distance Theory ­ If two objects at unequal distances from the observer form images of the same size on the retina, the more remote object must be larger ­ If the moon looks farther away when it is on the horizon than when it is higher in the sky, it should look larger Experiment ­ Pointed artificial moon apparatus at the horizon ­ Had observers viewing the “moon” through the hole in a sheet of cardboard that masked the terrain ­ The horizon moon looks no larger than the zenith moon ­ Pointed two of devices at the horizon: ­ A) Moon is viewed through a mask ­ B) Moon is seen over unobstructed terrain ­ Results: Moon B is 1.34 times larger than Moon A Terrain ­ Terrain is the key point for the moon illusion ­ The horizontal moon looks larger only because it is seen over terrain ­ It is possible to reverse the illusion by moving the terrain overheard with a mirror or prism ­ As expected, the illusion does reverse: the moon on an overheard horizon appears larger than the moon at a horizontal zenith, with a ratio of 1

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Chapter 9, Problem 58 is Solved
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Textbook: Algebra and Trigonometry
Edition: 3
Author: Cynthia Y. Young
ISBN: 9780470648032

Algebra and Trigonometry was written by and is associated to the ISBN: 9780470648032. This full solution covers the following key subjects: . This expansive textbook survival guide covers 13 chapters, and 10127 solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 3. The answer to “solve the system of linear equations.” is broken down into a number of easy to follow steps, and 6 words. Since the solution to 58 from 9 chapter was answered, more than 249 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 58 from chapter: 9 was answered by , our top Math solution expert on 01/04/18, 09:28PM.

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