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b). We all use antennas at our homes and a fact lies that antennas transmit some form of radiation. The radiation, however is not uniform in all directions. While conducting a research of the antenna radiation, a group of scientists modeled the intensity from a particular antenna by r=acos² 0. 1). Convert the polar equation to rectangular coordinates. (ii). Use Mathematica to plot the model for a = 4 and a = 6. (iii). Find the area of the geographical region between the two curves in past (ii). c). In polar coordinates, a polar function exists by the name of strophoid, with equation r=sec0-2cos 0. Consider the equation of this strophoid that is bounded as 1). Plot the strophoid. (11). Convert the strophoid equation to rectangular coordinates. (iii). Find the area enclosed by the loop.

b). We all use antennas at our homes and a fact lies that antennas transmit some form of radiation. The radiation, however is not uniform in all directions. While conducting a research of the antenna radiation, a group of scientists modeled the intensity from a particular antenna by r=acos² 0. 1). Convert the polar equation to rectangular coordinates. (ii). Use Mathematica to plot the model for a = 4 and a = 6. (iii). Find the area of the geographical region between the two curves in past (ii). c). In polar coordinates, a polar function exists by the name of strophoid, with equation r=sec0-2cos 0. Consider the equation of this strophoid that is bounded as 1). Plot the strophoid. (11). Convert the strophoid equation to rectangular coordinates. (iii). Find the area enclosed by the loop.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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b). We all use antennas at our homes and a fact lies that antennas transmit some form of radiation. The radiation, however is not uniform in all directions. While conducting a research of the antenna radiation, a group of scientists modeled the intensity from a particular antenna by r= acos? 0. (). Convert the polar equation to rectangular coordinates. (ii). Use Mathematica to plot the model for a = 4 and a = 6. (ii). Find the area of the geographical region between the two curves in past (ii). c). In polar coordinates, a polar function exists by the name of strophoid, with equationr=sec 0-2 cos 0. Consider the equation of this strophoid that is bounded as <0<: (1). Plot the strophoid. (ii). Convert the strophoid equation to rectangular coordinates. (iii). Find the area enclosed by the loop.
Transcribed Image Text:b). We all use antennas at our homes and a fact lies that antennas transmit some form of radiation. The radiation, however is not uniform in all directions. While conducting a research of the antenna radiation, a group of scientists modeled the intensity from a particular antenna by r= acos? 0. (). Convert the polar equation to rectangular coordinates. (ii). Use Mathematica to plot the model for a = 4 and a = 6. (ii). Find the area of the geographical region between the two curves in past (ii). c). In polar coordinates, a polar function exists by the name of strophoid, with equationr=sec 0-2 cos 0. Consider the equation of this strophoid that is bounded as <0<: (1). Plot the strophoid. (ii). Convert the strophoid equation to rectangular coordinates. (iii). Find the area enclosed by the loop.
Question 3 In robotics, a scheme, known as the Minimum Distance Technique (MDT) is used to avoid line obstacles. The MDT involves the calculation of the minimum distance from the robot to the line segment and the avoidance of the resultant point on the line segment. Avoidance of the closest point on the line at any time t20 essentially results in the avoidance of the entire line segment. Consider the Figure 3. Show that (a, - a,) (a, -a,)' +(b, –b.)° (b, – b,) (a, -a,)* +(b, –b,)* * 2= (x-a,)c+(y-b,)d where e= and d = Hint: 1. Determine the parametric equations of the line. 2. Find the Euclidean distance from the robot to the line segment. 3. Optimize the distance. Page | 3 JR. 入=0 入=1 (4,,b,) (a,.b,) (x, y) Mobile robot Figure 3: Schematic representation of a robot avoiding a line segment.
Transcribed Image Text:Question 3 In robotics, a scheme, known as the Minimum Distance Technique (MDT) is used to avoid line obstacles. The MDT involves the calculation of the minimum distance from the robot to the line segment and the avoidance of the resultant point on the line segment. Avoidance of the closest point on the line at any time t20 essentially results in the avoidance of the entire line segment. Consider the Figure 3. Show that (a, - a,) (a, -a,)' +(b, –b.)° (b, – b,) (a, -a,)* +(b, –b,)* * 2= (x-a,)c+(y-b,)d where e= and d = Hint: 1. Determine the parametric equations of the line. 2. Find the Euclidean distance from the robot to the line segment. 3. Optimize the distance. Page | 3 JR. 入=0 入=1 (4,,b,) (a,.b,) (x, y) Mobile robot Figure 3: Schematic representation of a robot avoiding a line segment.
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