Final answer:
The given region is the space between an upward-facing cone and a half-sphere resting above it. This region is defined by the equations x²+y²+z²=16 with z ≥ 0 for the half-sphere and 2z²=x²+y² for the cone. Graphically, it appears as a cone capped with a round, dome-like top.
Step-by-step explanation:
To graph the region bounded above by the half-sphere given by the equation x²+y²+z²=16 with z ≥ 0, and below by the cone given by the equation 2z²=x²+y², we visualize both three-dimensional shapes together.
The half-sphere represents the top portion of a sphere with a radius of 4, since the radius squared is equal to 16. The z-coordinate must be non-negative, which restricts the half-sphere to the upper half of the space (z ≥ 0).
Conversely, the cone's equation can be rewritten as z²=½(x²+y²), indicating that for every point (x, y) on the x-y plane, the height z of the cone at that point is the square root of half the distance squared from the origin to the point (x, y) on the plane. The vertex of the cone is at the origin (0, 0, 0), and it extends upwards, widening as it goes.
To find the intersection of these two surfaces, we can set the equations equal to each other: x²+y²+z² = 2z² simplifying to x²+y² = z², which we can see is a circle of radius z parallel to the xy-plane.
This circle is at the height where the half-sphere and the cone meet. On a graph, the region we are considering would appear as a conical shape cut off by the half-sphere, creating a rounded cap on the cone.