Question

Write the inequality to describe the region of the solid upper hemisphere of the sphere of radius 2, centered at the origin.

Short answer

The inequality to describe the region of the solid upper hemisphere of the sphere of radius 2, centered at the origin, is \({x^2} + {y^2} + {z^2} \le 4,z \ge 0\).

Step-by-step solution

Simplify the equation.

Formula used

\({(x - h)^2} + {(y - k)^2} + {(z - l)^2} = {r^2}(1)\)

Here,

\((h,k,l)\)is the sphere's center and

\(r\)is the radius of a sphere.

As the center of the sphere is at the origin, and its radius is 2, the center \(C(h,k,l)\) is \((0,0,0)\) and \(r\) is 2.

Complete the equation by substituting the values.

In equation (1), substitute 0 for \(h\),\(0\)for \(k\), and 0 for \(l\) and 2 for \(r\).

\(\begin{aligned}{l}{(x - 0)^2} + {(y - 0)^2} + {(z - 0)^2} = {2^2}\\{x^2} + {x^2} + {z^2} = 4\end{aligned}\)

Therefore, the equation \({x^2} + {x^2} + {z^2} = 4\) represents a sphere with radius \(2\) centered at the origin.

The equation to describe the region that consists of all points lie on the sphere with radius \(2\),centered at the origin is \({x^2} + {x^2} + {z^2} = 4\).

The inequality is used to characterize the portion of a solid sphere of radius 2 that is centered at the origin which is \({x^2} + {y^2} + {z^2} \le 4\). As the region required to describe is an upper hemisphere, the \(z\)-coordinate is restricted only to positive values (non-negative values of a z-coordinate).

Therefore, the inequality to describe the region of an upper hemisphere is \(z \ge 0\).

Thus, the inequality to describe the region of a solid upper hemisphere of the sphere of radius 2, centered at the origin, is \({x^2} + {y^2} + {z^2} \le 4,z \ge 0\).