Elimination. Previously, we learned that the parametric equation \(\displaystyle{x}={\cos{{8}}}{t}\) and \(\displaystyle{y}={\sin{{8}}}{t}\) lie on a circle on the xy-plane in \(\displaystyle{R}^{{{2}}}\). In \(\displaystyle{R}^{{{3}}}\), these parametric equations lie on a cylinder pointing to the z-axis. Only IV and VI fits this criterion as they are circular and the circles are oriented along the xy-plane. We have answered VI earlier so this must be IV. Let's check!

Step 2

Further elimination and conclusion.

Between the two equations, observe that the 2-component:

\(\displaystyle{z}={e}^{{{2}{t}}}\) is oscillating at an increasing amount. z is defined for all t and is not bounded by any values of t. Graph VI seems to be asymptotic to 0 whereas graph IV has circles that have an increasing period between them. Therefore, this equation must be describing graph IV.

Answer