The problem is to identify and describe the graph of the equation $z = \sqrt{16 - x^2 - y^2}$ in three-dimensional space.

1. Problem Description

The problem is to identify and describe the graph of the equation $z = \sqrt{16 - x^2 - y^2}$ in three-dimensional space.

2. Solution Steps

The given equation is $z = \sqrt{16 - x^2 - y^2}$.

Squaring both sides of the equation, we get

$z^2 = 16 - x^2 - y^2$.

Rearranging the terms, we obtain

$x^2 + y^2 + z^2 = 16$.

This is the equation of a sphere centered at the origin $(0, 0, 0)$ with radius $r$ such that $r^2 = 16$, which means $r = 4$.

However, since $z = \sqrt{16 - x^2 - y^2}$, we have $z \ge 0$.

Therefore, the graph is the upper hemisphere of the sphere $x^2 + y^2 + z^2 = 16$.

3. Final Answer

The graph of the equation $z = \sqrt{16 - x^2 - y^2}$ is the upper hemisphere of a sphere with radius 4 centered at the origin.