Question

Write inequalities to describe the solid left (x < 0 is left) hemisphere of a sphere of radius 4 centered at the origin

Answer 1

The equation of the sphere centered at 0, and radius 4 is:


`\displaystyle{ x^2+y^2+z^2=4^2`,

note that this equation describes exactly the points of the surface of the square. That is, this is an EMPTY sphere.

The solid sphere, that is the points on the surface and all points in the inside, are given by :


`\displaystyle{ x^2+y^2+z^2 \leq 4^2`


since we want the left part of the solid part, picture 2, we add the condition x<0,


thus "the solid left (x < 0 is left) hemisphere of a sphere of radius 4 centered at the origin" is given by the system of inequalities:


`i) \displaystyle{ x^2+y^2+z^2 \leq 4^2`

`ii)x\ \textless \ 0`