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) $\mathbf { r } ( t ) = ( 3 + 3 \sin t ) \mathbf { i } + ( 3 - 3 \cos t ) \mathbf { j } + \sqrt { 6 } \cos t \mathbf { k }$

Question

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) $\mathbf { r } ( t ) = ( 3 + 3 \sin t ) \mathbf { i } + ( 3 - 3 \cos t ) \mathbf { j } + \sqrt { 6 } \cos t \mathbf { k }$

Quizplus · Accepted Answer

Given $x = 3 + 3\sin t$, and $x + y = 6$, solving for $y$ gives $y = 3 - 3\sin t$. To satisfy $x^2 + y^2 + z^2 = 36$, substituting $x$ and $y$ and solving for $z$ yields $z = 3\sqrt{2}\cos t$. Thus, the correct vector-valued function is $\mathbf{r}(t) = (3 + 3\sin t)\mathbf{i} + (3 - 3\sin t)\mathbf{j} + 3\sqrt{2}\cos t\mathbf{k}$.

Find a Vector-Valued Function, Using the Given Parameter, to Represent Surfaces Parameter x2+y2+z2=36,x+y=6x=3+3sin⁡t\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\x^{2}+y^{2}+z^{2}=36, x+y=6 & x=3+3 \sin t\end{array} Surfaces x2+y2+z2=36,x+y=6​ Parameter x=3+3sint​

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