3.1: Vector Functions and Space Curve WEBASSIGN?

Question

x = (1 + cos 12t) cos t
y = (1 + cos 12t) sin t
z = 1 + cos 12t

Explain the appearance of the graph by showing that it lies on a cone.

Answer 1

Note that showing that x,y,z satisfy

x²+y²=z² establishes that it lies on a cone, since this is the equation of a cone.

 

Next, z is oscillating between 2 and 0 twelve times for every full 2kpi..2(k+1)pi rotation of t. For example, z oscillates from 2 to 0 twelve times between t=0 and t=2pi, again from 2pi to 4pi, etc. Meanwhile, x and y are tracing out a circle on the cone at height z above the xy-plane, with one rotation counterclockwise from above for every full 2kpi..2(k+1)pi rotation of t.

 

Graph 1 + cos 12t in desmos.com, and graph cos t, and imagine the oscillations up and down wrapping around the top half of the cone one time for every twelve oscillations. Note that they meet at the origin twelve times for each cycle around the cone as well.

 

I hope that helps.