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

1. Problem Description

We are asked 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 $z = \sqrt{16 - x^2 - y^2}$. We can square both sides to eliminate the square root, yielding $z^2 = 16 - x^2 - y^2$. Rearranging the terms, we have $x^2 + y^2 + z^2 = 16$. Since $z = \sqrt{16 - x^2 - y^2}$, we have $z \ge 0$.

The equation $x^2 + y^2 + z^2 = 16$ represents a sphere centered at the origin with radius $\sqrt{16} = 4$. Since $z \ge 0$, we only have the upper half of the sphere. Thus the graph is the upper hemisphere of radius 4 centered at the origin.

3. Final Answer

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