Problem 95 - Precalculus with Calculus Previews

Question

Describe geometrically all points in 3-space whose coordinates satisfy the given condition(s). $$ 1 \leq x^{2}+y^{2}+z^{2} \leq 9 $$

Step-by-Step Solution

Answer

The points form a hollow spherical shell between radii 1 and 3, centered at the origin.

Step 1: Understand the Concept of a Sphere

In three-dimensional space, the equation \( x^2 + y^2 + z^2 = r^2 \) represents a sphere with its center at the origin \((0,0,0)\) and radius \( r \). In this exercise, we will explore how this concept applies to the given inequalities.

Step 2: Analyze Lower Bound Inequality

The inequality \( x^2 + y^2 + z^2 \geq 1 \) indicates all points are on or outside a sphere of radius 1, centered at the origin. This represents a solid sphere's outer boundary and all points beyond it.

Step 3: Analyze Upper Bound Inequality

The inequality \( x^2 + y^2 + z^2 \leq 9 \) indicates all points are on or inside a sphere of radius 3, also centered at the origin. This represents a solid sphere with a maximum extent of radius 3.

Step 4: Combine Inequalities

When combining both inequalities, \( 1 \leq x^2 + y^2 + z^2 \leq 9 \), we find the solution includes points that are between two concentric spheres: outside or on the sphere of radius 1, but inside or on the sphere of radius 3. This describes a hollow spherical shell.

Key Concepts

Sphere

In three-dimensional geometry, a sphere is a perfectly symmetrical object. Every point on the surface of a sphere is at an equal distance from its center. This distance is known as the radius. The equation of a sphere centered at the origin can be written as \( x^2 + y^2 + z^2 = r^2 \), where \( r \) is the radius.Spheres have fascinating properties, including:

• Symmetry: Spheres are symmetric in all three dimensions.
• Surface Area: The surface area of a sphere is calculated by \( 4\pi r^2 \).
• Volume: Its volume is \( \frac{4}{3}\pi r^3 \).

Understanding the sphere is crucial because it forms the core around the concepts explored in inequalities in geometry and concentric spheres.

Inequalities in Geometry

Inequalities in geometry often describe regions or sets of points in space. They can define borders, boundaries, or entire areas. When dealing with spheres, inequalities can indicate whether points lie inside, on, or outside the sphere.For example:

• \( x^2 + y^2 + z^2 \leq r^2 \): This describes all points either inside or on the surface of the sphere.
• \( x^2 + y^2 + z^2 \geq r^2 \): This describes points that are on or outside the sphere.

In the original problem, we were given a compound inequality: \[1 \leq x^2 + y^2 + z^2 \leq 9\]This means the points are neither too close to the origin (closer than radius 1) nor too far away (farther than radius 3). This leaves a shell of space that is shaped like a hollow sphere.

Concentric Spheres

Concentric spheres share the same center but vary in radius. In the previous exercise, we explored two concentric spheres:

• The smaller sphere with a radius of 1, described by \( x^2 + y^2 + z^2 = 1 \).
• The larger sphere with a radius of 3, described by \( x^2 + y^2 + z^2 = 9 \).

When working with concentric spheres, you get structures known as spherical shells. The solution to the problem illustrates the concept of a spherical shell, which is a region between two concentric spheres. It is like having a solid sphere subtracted by a smaller solid sphere inside it.In practical terms, visualizing concentric spheres can be likened to a Russian doll, where a smaller sphere sits perfectly within a larger one, leaving space in between.