Ever wondered how fast a hula hoop is moving when you give it a push and it rolls along the oor? Consider a special hula hoop that has one very narrow black hand painted around it. If the hula hoop is positioned vertically on the oor so that the black band is touching the oor, when the hula hoop is given a push, it begins to travel upright along the oor. The path traced out by the black band is given by the position vector r= (t sin t) i + (l - cos t) j. a) Graph the vector function that represents the path traced out by the black band either with a graphing utility or sketch it by hand. Make certain to include your Window settings and provide justication for your setting selections. b) What are the vector functions that represent the velocity and acceleration? c) Find the maximum and minimum values of the velocity vector analytically. d) Find the maximum and minimum values of the acceleration vector analytically. e) Provide a graphical representation of the velocity and acceleration vectors. Do they conrm your answers to c and d? Suppose a friend wants to join in the fun and has a smaller hula hoop with a similar narrow black hand painted around it. The black band hits the oor twice as many times as yours. a) What is the vector function that models this behavior? b) Compare the paths traced out by each hula hoop. What are their similarities and differences? Make certain to include your Window settings and provide justication for your setting selections