The graph below is most likely the graph of which of the following equations?
A) $\mathbf { r } ( t ) = 2 t \cos ( t ) \mathbf { i } + 2 t \sin t \mathbf { j } + 2 \mathbf { k } , \quad 0 \leq t \leq 2 \pi$
B) $\mathbf { r } ( t ) = 2 \sin ( t ) \mathbf { i } + 2 \cos t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$
C) $\mathbf { r } ( t ) = 2 \cos ( t ) \mathbf { i } + 2 t \sin t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$
D) $\mathbf { r } ( t ) = 2 t \cos ( t ) \mathbf { i } + 2 \sin t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$
E) $\mathbf { r } ( t ) = 2 \sin ( t ) \mathbf { i } + 2 \sin t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$

Question

The graph below is most likely the graph of which of the following equations? 
A) $\mathbf { r } ( t ) = 2 t \cos ( t ) \mathbf { i } + 2 t \sin t \mathbf { j } + 2 \mathbf { k } , \quad 0 \leq t \leq 2 \pi$ 
B) $\mathbf { r } ( t ) = 2 \sin ( t ) \mathbf { i } + 2 \cos t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$ 
C) $\mathbf { r } ( t ) = 2 \cos ( t ) \mathbf { i } + 2 t \sin t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$ 
D) $\mathbf { r } ( t ) = 2 t \cos ( t ) \mathbf { i } + 2 \sin t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$ 
E) $\mathbf { r } ( t ) = 2 \sin ( t ) \mathbf { i } + 2 \sin t \mathbf { j } + 2 \mathbf { k } , 0 \leq t \leq 2 \pi$

Quizplus · Accepted Answer

The graph suggests a circular motion around the z-axis with a constant radius of 2. Starting at the point (0,2,2), it goes through the xy-plane to (-2,0,2), then to (0,-2,2), then to (2,0,2), and finally back to the starting point. This is the motion of a circle in the xy-plane with center at the origin and radius 2. Therefore, we can eliminate choices A, C, D, and E. The only choice that matches this description is B.

The Graph Below Is Most Likely the Graph of Which r(t)=2tcos⁡(t)i+2tsin⁡tj+2k,0≤t≤2π\mathbf { r } ( t ) = 2 t \cos ( t ) \mathbf { i } + 2 t \sin t \mathbf { j } + 2 \mathbf { k } , \quad 0 \leq t \leq 2 \pir(t)=2tcos(t)i+2tsintj+2k,0≤t≤2π

The Graph Below Is Most Likely the Graph of Which $\mathbf { r } ( t ) = 2 t \cos ( t ) \mathbf { i } + 2 t \sin t \mathbf { j } + 2 \mathbf { k } , \quad 0 \leq t \leq 2 \pi$