Find a Vector-Valued Function, Using the Given Parameter, to Represent $\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\x^{2}+y^{2}+z^{2}=36, x+y=6 & x=3+3 \sin t\end{array}$

Multiple Choice

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\x^{2}+y^{2}+z^{2}=36, x+y=6 & x=3+3 \sin t\end{array}$

A) $\mathbf { r } ( t ) = ( 3 + 3 \sin t ) \mathbf { i } + ( 3 - 3 \sin t ) \mathbf { j } + 3 \sqrt { 2 } \sin t \mathbf { k }$
B) $\mathbf { r } ( t ) = ( 3 + 3 \sin t ) \mathbf { i } + ( 3 - 3 \sin t ) \mathbf { j } + 3 \sqrt { 2 } \cos t \mathbf { k }$
C) $\mathbf { r } ( t ) = ( 3 + 3 \sin t ) \mathbf { i } + ( 3 + 3 \sin t ) \mathbf { j } + 3 \sqrt { 2 } \cos t \mathbf { k }$
D) $\mathbf { r } ( t ) = ( 3 + 3 \sin t ) \mathbf { i } + ( 3 - 3 \cos t ) \mathbf { j } + 3 \sqrt { 2 } \cos t \mathbf { k }$
E) $\mathbf { r } ( t ) = ( 3 + 3 \sin t ) \mathbf { i } + ( 3 - 3 \cos t ) \mathbf { j } + \sqrt { 6 } \cos t \mathbf { k }$