Question

How to write an inequality for top hemisphere of sphere

Answer 1

The correct inequality for this region is z ≥ 0 and x^2 + y^2 + z^2 ≤ 64, which describes the solid upper hemisphere of the sphere with radius 8 centered at the origin.

Step-by-step explanation:

To write an inequality to describe the region of a solid upper hemisphere, we need to consider both the radius of the sphere and the fact that it includes only the upper half (where z is positive). A sphere with radius 8 centered at the origin has the equation x2 + y2 + z2 = 82 or x2 + y2 + z2 = 64. Since we are only interested in the solid upper hemisphere, we also need to include z ≥ 0. Combining these gives us the inequality z ≥ 0, x2 + y2 + z2 ≤ 64, which corresponds to option 3 from your choices.