An object is moving in the plane with parametric equations: x(t) = cos(?t),y(t)

Question

An object is moving in the plane with parametric equations: x(t) = cos(?t),y(t) = 3sin(?t). The path traced out is an ellipse, as pictured. The location of the object at time t seconds is given by P(t)=(x(t),y(t)), where we assume t is non-negative and in units of "seconds". Assume units on the axes are "feet".

P(t -2 -2 (a) The initial coordinates of the object (i.e. at time t0) are (b) The location at time t-5/6 is (c) How many seconds does it take the object to go around the elliptical path exactly once? (d) The first time the y-coordinate of the object is -1 is seconds

Explanation / Answer

a) The initial coordinates of the object at time t=0 is given by

(cos(0),3sin(0)) = (1,0), hence the initial coordinate is equal to (1,0)

b) The location at time t=5pi/6

(cos(5pi/6),3sin(5pi/6) = (-0.866,1.5)

c) The time taken for one complete revolution of cos and sin is equal to 2pi

pi * t = 2pi

t = 2

Hence the time t=2 seconds

d) the y-coordinate will be -1, when

3sin(pi*t) = -1

sin(pi*t) = -1/3

pi * t = 199.45 * pi/180

t = 1.108 seconds

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