Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function $\mathbf { r } ( u , v ) = 9 \cos v \cos u \mathbf { i } + 9 \cos v \sin u \mathbf { j } + 6 \sin v \mathbf { k }$.
A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 6 } + \frac { z ^ { 2 } } { 6 } = 1$
B) $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 36 } + \frac { z ^ { 2 } } { 36 } = 1$
C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 81 } + \frac { z ^ { 2 } } { 81 } = 1$
D) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 9 } + \frac { z ^ { 2 } } { 6 } = 1$
E)$\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 81 } + \frac { z ^ { 2 } } { 36 } = 1$

Question

Quizplus · Accepted Answer

The given vector function represents an ellipsoid. By substituting $x = 9 \cos v \cos u$, $y = 9 \cos v \sin u$, and $z = 6 \sin v$, we can eliminate the parameters $u$ and $v$ to get the equation $\frac{x^2}{81} + \frac{y^2}{81} + \frac{z^2}{36} = 1$.

Find the Rectangular Equation for the Surface by Eliminating the Parameters