Find a parametric representation for the part of the elliptic paraboloid $x + y ^ { 2 } + 10 z ^ { 2 } = 5$ that lies in front of the plane $x = 0$.
A) $x = x , y = \sqrt { 5 - x - 10 z ^ { 2 } } , z = z$
B) $x = 5 - y ^ { 2 } - 10 z ^ { 2 } , y = y , z = y , y ^ { 2 } + 10 z ^ { 2 } \leq 5$
C) $x = 5 - y ^ { 2 } - 10 z ^ { 2 } , y = y , z = y , 0 \leq y ^ { 2 } + 10 z ^ { 2 } \leq 3$
D) $x = x , y = \pm \sqrt { 5 - x - 10 z ^ { 2 } } , z = z$
E) $x = 5 - y ^ { 2 } - 10 z ^ { 2 } , \mathrm { y } = y , z = y , y ^ { 2 } + 10 z ^ { 2 } \geq 5$

Question

Quizplus · Accepted Answer

We can solve for $z^2$ in terms of $x$ and $y$ by rearranging the equation given as: $z^2 = \dfrac{5-x-y^2}{10}$. Since we want the part of the elliptic paraboloid that lies in front of the plane $x=0$, we know that $x\geq 0$ and therefore $z^2 \geq 0$. Thus, we have $\dfrac{5-x-y^2}{10} \geq 0 \implies y^2 \leq 5-x$. This gives us the parametric representation $x=x, y=\pm\sqrt{5-x}, z=z$. Therefore, the answer is (D).

Find a Parametric Representation for the Part of the Elliptic x+y2+10z2=5x + y ^ { 2 } + 10 z ^ { 2 } = 5x+y2+10z2=5

Find a Parametric Representation for the Part of the Elliptic $x + y ^ { 2 } + 10 z ^ { 2 } = 5$