The rectangular equation for the parametric surfaceand $x ( u , v ) = 3 \cos u$ is

A) $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 4$
B) $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$
C) $x ^ { 2 } + y ^ { 2 } = 9$
D) $x ^ { 2 } - y ^ { 2 } = 9$
E) $- x ^ { 2 } + y ^ { 2 } = 9$

Question

The rectangular equation for the parametric surfaceand $x ( u , v ) = 3 \cos u$ is

A) $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 4$
B) $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$
C) $x ^ { 2 } + y ^ { 2 } = 9$
D) $x ^ { 2 } - y ^ { 2 } = 9$
E) $- x ^ { 2 } + y ^ { 2 } = 9$

Quizplus · Accepted Answer

Given $x(u, v) = 3\cos u$, squaring both sides gives $x^2 = 9\cos^2 u$. Since the equation does not involve $v$ or $z$, and there's no information about how $y$ is defined in terms of $u$ and $v$, the most logical conclusion is that $y$ is not constrained by $u$ or $v$, implying $x^2 + y^2 = 9$ is the correct rectangular equation, which represents a cylinder in three dimensions.

The Rectangular Equation for the Parametric Surfaceand x(u,v)=3cos⁡ux ( u , v ) = 3 \cos ux(u,v)=3cosu

The Rectangular Equation for the Parametric Surfaceand $x ( u , v ) = 3 \cos u$