1. The problem is to identify the geometric objects represented by the given equations and describe their orientation relative to the coordinate axes. 2. (a) Equation: $z = -1$ - This represents a plane parallel to the $xy$-plane, located at $z = -1$. - It is oriented horizontally, one unit below the $xy$-plane. 3. (b) Equations: $y = -1$, $z = 2$ - These represent two planes intersecting along a line parallel to the $x$-axis. - The line of intersection is at $y = -1$, $z = 2$, extending along $x$. 4. (c) Equation: $x = 2$, with $-2 \leq y \leq 2$, $-1 \leq z \leq 3$ - This is a plane parallel to the $yz$-plane, located at $x = 2$. - The region is bounded in $y$ and $z$ as given. 5. (d) Equation: $x^2 + y^2 = 9$ - This is a cylinder with radius 3, axis along the $z$-axis. - It extends infinitely in $z$ direction. 6. (e) Equations: $x^2 + y^2 = 4$, $z = y$ - This is a circular cylinder of radius 2 oriented along $z$ with the height $z$ equal to $y$. - The curve is a helix-like line on the cylinder. 7. (f) Equations: $x^2 + y^2 + z^2 = 16$, $x = 0$ - This is a sphere of radius 4 centered at origin. - The condition $x=0$ restricts to the $yz$-plane slice of the sphere, a circle of radius 4. 8. (g) Inequality: $x^2 + y^2 + z^2 \leq 9$ - This represents a solid sphere of radius 3 centered at the origin. 9. (h) Inequalities: $x^2 + y^2 \leq 4$, $0 \leq z \leq 3$ - This is a solid cylinder of radius 2 along the $z$-axis, bounded between $z=0$ and $z=3$. 10. (i) Equation: $z = y^2$ - This is a parabolic cylinder opening upward along the $z$-axis. - It extends infinitely in the $x$-direction. "slug": "3d geometric objects", "subject": "multivariable calculus", "desmos": {"latex": "", "features": {"intercepts": true, "extrema": true}}, "q_count": 9