Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}
\text { Surfaces } & \text { Parameter } \
x^{2}+y^{2}=25, z=x^{2} & x=5 \sin 2 \pi t
\end{array}$

A) $\mathbf { r } ( t ) = 25 \sin 2 \pi t \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } - 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$
B) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { j } + 5 \cos 2 \pi t \mathbf { i } + 25 \sin ^ { 2 } 2 \pi \mathbf { k }$
C) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } - 25 \sin ^ { 2 } 2 \pi \mathbf { k }$
D) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } + 25 \sin ^ { 2 } 2 \pi t \mathbf { k }$
E) $\mathbf { r } ( t ) = 25 \sin 2 \pi \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } + 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$

Question

Find a vector-valued function, using the given parameter, to represent the intersection of the surfaces given below. $\begin{array}{ll}
\text { Surfaces } & \text { Parameter } \
x^{2}+y^{2}=25, z=x^{2} & x=5 \sin 2 \pi t
\end{array}$

A) $\mathbf { r } ( t ) = 25 \sin 2 \pi t \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } - 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$
B) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { j } + 5 \cos 2 \pi t \mathbf { i } + 25 \sin ^ { 2 } 2 \pi \mathbf { k }$
C) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } - 25 \sin ^ { 2 } 2 \pi \mathbf { k }$
D) $\mathbf { r } ( t ) = 5 \sin 2 \pi t \mathbf { i } + 5 \cos 2 \pi t \mathbf { j } + 25 \sin ^ { 2 } 2 \pi t \mathbf { k }$
E) $\mathbf { r } ( t ) = 25 \sin 2 \pi \mathbf { i } + 25 \cos 2 \pi t \mathbf { j } + 5 \sin ^ { 2 } 2 \pi t \mathbf { k }$

Quizplus · Accepted Answer

The intersection involves a cylinder of radius 5 centered around the $z$-axis and a sphere of radius 25 centered at the origin. It is clear that the intersection is symmetric with respect to the $z$-axis, so we want a vector-valued function with no $z$-component. This eliminates choices A, B, and E.

To ensure that the vector-valued function stays in the cylinder of radius 5, we want the $x$ and $y$ components to have magnitude no greater than 5. This eliminates choice C.

Thus, the only remaining option is D. We can check that this vector-valued function satisfies the necessary conditions and indeed represents the intersection of the surfaces.

Find a Vector-Valued Function, Using the Given Parameter, to Represent Surfaces Parameter x2+y2=25,z=x2x=5sin⁡2πt\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\x^{2}+y^{2}=25, z=x^{2} & x=5 \sin 2 \pi t\end{array} Surfaces x2+y2=25,z=x2​ Parameter x=5sin2πt​

Find a Vector-Valued Function, Using the Given Parameter, to Represent $\begin{array}{ll}\text { Surfaces } & \text { Parameter } \\x^{2}+y^{2}=25, z=x^{2} & x=5 \sin 2 \pi t\end{array}$