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入=0 入=1 (a,,b,) (a,.b.) (x, y) Mobile robot Figure 3: Schematic representation of a robot avoiding a line segment. Question 4 a). Use a double integral to find the volume in the first ocyant bounded by the coordinate planes, the plane y=4, and the plane + =1. 35 b). Find the area in the first quadrant that is inside the circle r=4 sin 0 and outside the lemniscate r? = 8cos 20.

入=0 入=1 (a,,b,) (a,.b.) (x, y) Mobile robot Figure 3: Schematic representation of a robot avoiding a line segment. Question 4 a). Use a double integral to find the volume in the first ocyant bounded by the coordinate planes, the plane y=4, and the plane + =1. 35 b). Find the area in the first quadrant that is inside the circle r=4 sin 0 and outside the lemniscate r? = 8cos 20.

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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c). As a result of the water crisis in Lasa, the Lasa Department of Water, with the assistance of their engineers and accountants decided to build a water line that has to be cost effective. The water line Page | 2 JR will be built from point A to D, as shown in Figure 2. The cost differs, given as such: 3k from A to B, 2k from B to C, and k from C to D, where the cost per kilometer is in dollars. The accountants worked together with the engineers to minimize the total cost. Find x and y such that the total cost C will be minimized. 2km ! B Ikm D 10km Figure 2: The water line plan. d). Use Lagrange multipliers to find the highest and the lowest points on the curve of intersection of z = x² +4y and x² + y =1. 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 (b, –b,) (a, -a,)° +(b, –b,)* (а, -а,) 1=(x-a,)c+(y-b)d where c= and d = (a, -a,)° +(b, –b,)² 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. 2=0 2=1 (a,.b) (a,.b.) (x, y) Mobile robot Figure 3: Schematic representation of a robot avoiding a line segment. Question 4 а). Use a double integral to find the volume in the first ocyant bounded by the coordinate planes, the х г +==1. 3 5 plane y=4, and the plane b). Find the area in the first quadrant that is inside the circle r= 4 sin 0 and outside the lemniscate p2 = 8cos 20.
Transcribed Image Text:c). As a result of the water crisis in Lasa, the Lasa Department of Water, with the assistance of their engineers and accountants decided to build a water line that has to be cost effective. The water line Page | 2 JR will be built from point A to D, as shown in Figure 2. The cost differs, given as such: 3k from A to B, 2k from B to C, and k from C to D, where the cost per kilometer is in dollars. The accountants worked together with the engineers to minimize the total cost. Find x and y such that the total cost C will be minimized. 2km ! B Ikm D 10km Figure 2: The water line plan. d). Use Lagrange multipliers to find the highest and the lowest points on the curve of intersection of z = x² +4y and x² + y =1. 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 (b, –b,) (a, -a,)° +(b, –b,)* (а, -а,) 1=(x-a,)c+(y-b)d where c= and d = (a, -a,)° +(b, –b,)² 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. 2=0 2=1 (a,.b) (a,.b.) (x, y) Mobile robot Figure 3: Schematic representation of a robot avoiding a line segment. Question 4 а). Use a double integral to find the volume in the first ocyant bounded by the coordinate planes, the х г +==1. 3 5 plane y=4, and the plane b). Find the area in the first quadrant that is inside the circle r= 4 sin 0 and outside the lemniscate p2 = 8cos 20.
Question 1 a). While playing at the Nawaka Sevens, four of the rugby players are observed to be located at the corners such that it makes a perfect square with sides of length d. The coach has brought about a new technique to train players and it is observed that all the players move counterclockwise at the same speed towards the next rugby player, as shown in the Figure 1 below. Find the polar equation of a rugby player's path as it spirals towards the centre of the square. d Figure 1: The movement of the rugby players of the square pitch. Page | 1 JR. 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. (i). 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 equation r= sec0- 2 cos 0. Consider the equation of this strophoid that is bounded as 2 (1). Plot the strophoid. (ii). Convert the strophoid equation to rectangular coordinates. (iii).Find the area enclosed by the loop. Question 2 а). Let f(x, y), where x= g (t) and y =h(t). d ( dz dx, dz dy and ôʻz dx _ d°z dy 1). Show that + it dt Əx. ôx² dt ðyôx dt dt ây дхду dt Ôy² dt d'z (ii). Find dr? b). du Len u(x, y) and v(x, y). The Cauchy-Riemann differential equations are given as and ôu ). Verify that the Cauchy-Riemann differential equations can be written in polar coordinate form as 1 ôv dv 1 du and r d0 %3D r d0 (11). Using the functions u = In Jx + y² and v= tan E, verify the two forms of the Cauchy- Riemann differential equations.
Transcribed Image Text:Question 1 a). While playing at the Nawaka Sevens, four of the rugby players are observed to be located at the corners such that it makes a perfect square with sides of length d. The coach has brought about a new technique to train players and it is observed that all the players move counterclockwise at the same speed towards the next rugby player, as shown in the Figure 1 below. Find the polar equation of a rugby player's path as it spirals towards the centre of the square. d Figure 1: The movement of the rugby players of the square pitch. Page | 1 JR. 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. (i). 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 equation r= sec0- 2 cos 0. Consider the equation of this strophoid that is bounded as 2 (1). Plot the strophoid. (ii). Convert the strophoid equation to rectangular coordinates. (iii).Find the area enclosed by the loop. Question 2 а). Let f(x, y), where x= g (t) and y =h(t). d ( dz dx, dz dy and ôʻz dx _ d°z dy 1). Show that + it dt Əx. ôx² dt ðyôx dt dt ây дхду dt Ôy² dt d'z (ii). Find dr? b). du Len u(x, y) and v(x, y). The Cauchy-Riemann differential equations are given as and ôu ). Verify that the Cauchy-Riemann differential equations can be written in polar coordinate form as 1 ôv dv 1 du and r d0 %3D r d0 (11). Using the functions u = In Jx + y² and v= tan E, verify the two forms of the Cauchy- Riemann differential equations.
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