Question

Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. 21.\(x = cos\;8t,\;y = sin\;8t,\;z = {e^{0.8t}},\;t \ge 0\).

Short answer

The answer is graph no IV

Step-by-step solution

Step 1: Elimination

Previously we learned that the parametric equation \(x = \cos \;8t\;\& \;y = \sin \;8t\) lie on a circle on the xy plane in \({\mathbb{R}^2}\). In \({\mathbb{R}^3}\), these parametric equations lie on a cylinder pointing to 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

So between the two equations observe that the z component: \(z = {e^{2t}}\)

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 zero whereas graph IV has circles that have an increasing period between them. Therefore, this equation must be describing graph IV.