Question

Write the inequality to describe the region that consists of all points between the spheres of radiiand \(R\) centered at the origin.

Short answer

The inequality to describe the region that consists of all points between the spheres of radii \(r\) and \(R\) centered at the origin is \({r^2} < {x^2} + {y^2} + {z^2} < {R^2}\).

Step-by-step solution

Simplify the equation.

\({R^3}\)is the three-dimensional coordinate system which contains\(x\),\(y\),and\(z\)-coordinates.

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, the center \(C(h,k,l)\) is \((0,0,0)\).

Complete the equation by substituting the values.

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

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

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

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

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

As the radius \(r\) is lesser than the radius \(R\), the lower limit of region is \(r\) and the upper limit is \(R\).

Thus, the inequality to describe the region that consists of all points between the spheres of radii \(r\) and \(R\) centered at the origin is \({r^2} < {x^2} + {y^2} + {z^2} < {R^2}\).