A particle's position in the plane t seconds after it starts moving is
x = 6 cos kt
y = 6 sin kt
for some constant k. If the particle moves a distance of 120 pi for every second, then what is k?
The distance traveled by the particle in t seconds is given by the perimeter of the ellipse it traces out. The semi-major and semi-minor axes of the ellipse are 6 and 6, respectively, so the distance traveled is 2π⋅6⋅6=72π.
Since the distance traveled is 120 pi for every second, we have k⋅72π=120π. Solving for k, we find k=35.