The "Little Old Lady from Pasadena\" drove her car 300 miles from Dallas to Lubbock at a rate of 80 miles per hour. Her good friend "Surfer Boy", drove his car from Dallas to Lubbock at a rate of 50 miles per hour (you know how laid back surfers are). The two leave at the same time and follow the same linear path from Dallas to Lubbock. a) Write parametric equations that express the "Little Old Lady's" position [distance from Dallas] as a function of time. b) Write parametric equations that express "Surfer Boy's position [distance from Dallas] as a function of time. c) Enter the equations into your calculator and Select appropriate Window * 5pones Settings for viewing below. d) After 2 hours, who is ahead and by how much? * f) If the \"Surfer Boy" (x:, y-) starts out 150 miles closer to Lubbock thanthe * 5poms Little Old Lady (x., v.), how will the original eguations change? g) Under these new conditions, who arrives first and how much sooner? * Paul, who is running in a Marathon [26.2188 miles], runs consistently at an average speed of 6 miles per hour. Two hours after he leaves home, on the day of the Marathon, his mother needs 1o give him a message. She sends his sister, Maria, on her bike to give him the message. Maria rides at an average speed of 15 mph. Write parametric equations to model this situation and determine from the graph when she will catch up to Paul. > a) Write parametric equations that express Paul's position [distance from * 5points home] as a function of time. Choose - b) Write parametric equations that express Maria's position [distance from * 5 points home] as a function of time. Choose - c) Enter the equations into your calculator and Select appropriate Window * 5 points Settings for viewing below. d) How long does it take Maria to catch up with Paul? [Remember: Maria * 5points starts two hours after Paul, at time T = 2] Choose - d) How far has Paul run, and Maria cycled, when she catches up to him? * 5 points Sherry leaves Columbus, Ohio on a Train to Dallas traveling at a constant rate of 50 mph. Five hours later, Larry leaves Dallas on a Bus to Columbus traveling at a constant rate of 70 mph. Sherry and Larry follow the same route [the train tracks and the highway are parallel] travelling in opposite directions without any stops. Columbus and Dallas are 950 miles apart. T o - a) Write parametric equations that express Sherry's position [distance from * 5 paints Columbus] as a function of time. Choose v b) Write parametric equations that express Larry's position [distance from * 5paints Columbus] as a function of time. [Remember: This time Larry is traveling in the opposite direction.] Choose - c) Enter the equations into your calculator and Select appropriate Window * 5 paoints Settings for viewing below. Choose - d) What is the value of T when Sherry arrives in Dallas, and how many * 5 points hours does her trip take? Choose - c) Enter the equations into your calculator and Select appropriate Window * 5 points Settings for viewing below. Choose - d) What is the value of T when Sherry arrives in Dallas, and how many * 5 points hours does her trip take? Choose - e) What is the value of T when Larry arrives in Columbus, and how many * 5points hours does his trip take? [Remember: Larry starts five hours after Sherry, attime T = 5] Choose - e) What is the value of T when Sherry and Larry pass by each other, like * 5 points ships in the night, travelling in opposite directions? Choose - f) How far has Sherry travelled when she and Larry pass each other? * 5 points Choose - g) How far has Larry travelled when he and Sherry pass each other? * 5 points Choose - A dart is thrown upward with an initial velocity of 58 ft/sec from an initial height of 4.5 feet and at an angle of elevation of 410. Consider the position of the dart at any time t, where t = 0 when the dart is thrown. Neglect air resistance. 58 ft/sec 58 sin 410 8 = 410 58 cos 410 4.5 ft X a) Write parametric equations to represent the 2-Dimensional Path of the *0/5 Dart. 1. x(t) = 58cos(41.)t, y(t) = 16t2 - 58sin(419)t + 4.5 2. x(t) = -16t2 + 58sin(41.)t + 4.5, y(t) = 58cos(41.)t X 3. x(t) = 58cos(41 )t, y(t) = -16t2 + 58sin(419)t + 4.5 4. x(t) = 58cos(41 )t + 4.5, y(t) = -16t2 + 58sin(410)t 5. x(t) = 58cos(41 )t, y(t) = -16t2 + 58sin(410)t - 4.5Forces with magnitudes of 2000 newtons and 900 newtons act on a machine part at angles of 10' and 85 respectively, with the x-axis. Find the direction and the magnitude of the resultant of these forces. 900 N 105 85 2000 N +10 Using the Law of Cosines, find the magnitude [length] of vector r. This will * 10 points be the net force acting on the machine. [Formulas for the Law of Cosines are shown below.] LAW of COSINES: Given AABC c2 = a2 + b2 - 2ab cos(C) a2 + b2 - c2 COS C = 2ab Choose Using the Law of Sines, find angle alpha and then add 10 to find the * 10 points direction angle, Or, for vector r. This will be the angle at which the net force is acting on the machine. [The Formula for the Law of Sines is shown below.] LAW of SINES: Given AABC sin A sin B sin C a b C ChooseMake a diagram that illustrates the velocity of an airplane heading Due East at 400 knots. Illustrate a wind velocity of 50 knots blowing toward the northeast at a Compass Direction of exactly 450 If the plane encounters this wind, illustrate its resultant velocity. Calculate the resultant speed and True Compass Direction of the plane. N Compass Direction of Airplane Course Course & Ground Speed 1350 Wind Speed: 50 knots Compass Direction: 45 W - Heading & Air Speed: 400 knots Using the Law of Cosines, find the magnitude [length] of vector r. This will * 10 points be the Ground Speed of the Airplane. [Formulas for the Law of Cosines are shown below.] LAW of COSINES: Given AABC c2 = a2 + b2 - 2ab cos(C) a2 + 62 - c2 COS C = 2ab Choose Using the Law of Sines, find the direction angle, Or, for vector r, then * 10 points subtract this angle from 90 to get the Compass Direction of the actual Course of the Airplane. [The Formula for the Law of Sines is shown below.] LAW of SINES: Given AABC sin A sin B sin C a b C ChooseAn airplane's Air Speed is 580 miles per hour, and its Heading is 58. The wind is from the southeast at 315" and has a velocity of 60 miles per hour. Draw a figure that gives a visual representation of the problem. Find the actual Ground Speed and True Compass Direction of the Airplane. Using the Law of Cosines, find the magnitude [length] of vector r. This will * 10 points be the Ground Speed of the Airplane. [Formulas for the Law of Cosines are shown below.] LAW of COSINES: Given AABC c? = a* + b* 2ab cos(C) Choose - Using the Law of Sines, find angle alpha, then subtract from 58 to get the * 10 points Compass Direction of the actual Course of the Airplane. [The Formula for the Law of Sines is shown below.] LAW of SINES: Given AABC sin A _sinB B sinC a b C Choose - A Motorboat is headed Due East, directly across a river at 5 m/S. The current of the river is 2 m/s downstream (Due South). Find the following: a) the resulting True Speed of the boat; b) the Compass Direction of the boat; and c) the distance downstream the boat will land on the shore if the river is 800 meters wide. [Note: These vectors form a right triangle so you can use Right Triangle Trig and the Pythagorean Theorem to work the problem!] N Compass Direction of Motorboat Course Heading & Speed: 5 m/s W - E Current: 2 m/s True Course & Speed a) Use the Pythagorean Theorem to find the magnitude [length] of vector * 10 points r. This will be the True Speed of the Motorboat. Choose b) Using the Tangent Function, find the direction angle, Or, for vector r, [as * 10 points drawn 0, is negative] then add the absolute value of 0, to 90 to get the Compass Direction of the actual Course of the Motorboat. Choose c) Use the figure below and the direction angle, Or, that you found above * 10 points for vector r, to determine how far downstream the Motorboat will land on the opposite shore if the river is 800 meters wide. N Compass Direction of Motorboat Course W Width of River: 800 meters E d= the Motorboat Downstream Distance River Current Carries S True Course & Distance Traveled ChooseFind the Dot Product of the following Two Vectors: (2, 3), (4, -5) * Dot Product of Two Vectors If u = (x1, y1) and v = (x2, yz), then u . v = x1X2 + y1)/2 Note: The dot product is a scalar. Choose Find the Angle Between the following Two Vectors: (2, 3), (4, -5) * Angle Between Two Vectors u . v COS 0 = for 0' S 0 X2 Choose v Find the value of "a" if the vectors (6, -8) and (4, a) are perpendicular. * Perpendicular Vectors Two vectors v and u are perpendicular if veu = Choose v Determine if the Vector (5, 2) and the Vector (-10, -4) are Parallel, Perpendicular, or Neither. Parallel Vectors For two vectors U = (x;,y,) and = (x5, y,), if 21 = f then the vectors are parallel. A Perpendicular Vectors Two vectors v and u are perpendicular if veud =0 Choose v Determine if the Vector (1, 3) and the Vector (-8, 5) are Parallel, Perpendicular, or Neither. Parallel Vectors For two vectors u = (x,y,) and v = (x3,,), if i' = f then the vectors are parallel. 1 2 Perpendicular Vectors Two vectors v and u are perpendicular if veil =0 Choose v Component Form and Magnitude of a Vector Given Two Points The Component Form of the vector with initial point A(a,,a,) and terminal point B(b,, b, ) is given by: AB = (by ay, b, a,) =7 The Magnitude (or length) of is given by [] = V(b ay)? + (b, a,)? Given Initial Point P(3, -9) and Terminal Point Q(0, -3), find the Component * 5 points Form of Vector PQ. Choose v Given Initial Point P(3, -9) and Terminal Point Q(0, -3), find the Magnitude * 5 points of Vector PQ. Choose v VECTOR EQUATION FOR AN OBJECT MOVING WITH CONSTANT RATE OF CHANGE (VELOCITY) (X, Y) = (x0,y0) + t(Ax, Ay) (xo, o) = Initial Position (Ax,Ay) = Velocity (Change per unit of time) |Velocity| = |(Ax, Ay)| = Speed Write a Vector Equation for a Moving Object with Constant Velocity = (3, * 7 points -1) and Initial Position at time t = 0 of (2, 3). Choose v Write Parametric Equations for the moving object described above. * 7 points Choose v What is the speed of the moving object described above. * 6 points Choose v VECTOR EQUATION FOR AN OBJECT MOVING WITH CONSTANT RATE OF CHANGE (VELOCITY) (X,Y) = (x0,y0) + t{Ax, Ay) (x0,o) = Initial Position (Ax, Ay) = Velocity (Change per unit of time) |Velocity| = |{Ax, Ay)| = Speed Write a Vector Equation for a Moving Object with Constant Velocity = (1, * 7 points -1) and Initial Position at time t = 0 of (1, -5). Choose - Write Parametric Equations for the moving object described above. * 7 points Choose - What is the speed of the moving object described above. * 6 points Choose v Write a Vector Equation for an Object Moving with Constant Velocity having an Initial Position at time t = 0 of (5, 6) and a Position attimet=3 of (-19, -12). Choose - Write Parametric Equations for the moving object described above. * Choose v What is the speed of the moving object described above. * Choose - VECTOR EQUAITION FOR AN OBJEC T MOVING WIIH CONSTANT RATE OF CHANGE (VELOCITY) (X, Y) = (x0,0) + t{Ax, Ay) (x0,yo) = Initial Position (Ax, Ay) = Velocity (Change per unit of time) |Velocity| = [{Ax, Ay)| = Speed Write a Vector Equation for an Object Moving with Constant Velocity * 7 points having an Initial Position at time t = 0 of (1, 4) and a Position at timet =4 of (21, -16). Choose v Write Parametric Equations for the moving object described above. * 7 points Choose v What is the speed of the moving object described above. * 6 points Choose - Write a Vector Equation for an Object Moving with Constant Velocity having an Initial Position at time t = 0 of (-2, 3) and a Position attimet =6 of (-38, 27). Choose v Write Parametric Equations for the moving object described above. * Choose v What is the speed of the moving object described above. * Choose v