Consider the curve C parametrized by x = cos(6t) and y = sin(6t), for 0 <= t <=
ID: 3284210 • Letter: C
Question
Consider the curve C parametrized by x = cos(6t) and y = sin(6t), for 0 <= t <= pi/6 and traversed by a particle moving along C with increasing values of t. Choose the correct answer below. (a) The particle travels clockwise, traversing 1/12 of the unit circle. (b) The particle travels counterclockwise, traversing 1/12 of the unit circle. (c) The particle travels clockwise, traversing 1/6 of the unit circle . (d) The particle travels counterclockwise, traversing 1/6 of the unit circle. (e) The particle travels clockwise, traversing 1/2 of the unit circle. (f) The particle travels counterclockwise, traversing 1/2 of the unit circle. (g)The particle travels clockwise, 6 times around the unit circle. (h)The particle travels counterclockwise, 6 times around the unit circle.Explanation / Answer
the equation of the path satisfies x^2 + y^2 = 1 . so we know it travels in a unit circle. at time t = 0 the location is (1,0) on the x-y plane at half the time that is at t= pi/12 we have x = cos(6*pi/12) = cos(pi/2) =0 and y =sin(6*pi/12) = sin(pi/2) =1 so its at (0,1) at the end its at x = cos(6*pi/6) = cos(pi) = -1 and y =sin(6*pi/6) = sin(pi) =0 so its at (-1,0) so knowing three points we know it travels in the counter clockwise also it travels half the unit circle because the starting and end points are diametrically opposite , with their midpoint at the origin which is incidentally the centre of the circle . Hence the answer is f.The particle travels counterclockwise, traversing 1/2 of the unit circle.