Problem 39 - Precalculus with Calculus Previews

Question

Sketch the graph of the given equation. $$ x^{2}+y^{2}+z^{2}=9 $$

Step-by-Step Solution

Answer

The equation represents a sphere with center at the origin and radius 3.

Step 1: Identify the Type of Equation

The given equation is \(x^2 + y^2 + z^2 = 9\). This is an equation of a sphere.

Step 2: Compare to Standard Form

The standard form of a sphere's equation is \(x^2 + y^2 + z^2 = r^2\). This tells us that the sphere is centered at the origin \((0, 0, 0)\) and has a radius \(r = 3\), since \(r^2 = 9\).

Step 3: Visualize the Sphere

In three-dimensional space, this sphere is centered at the origin (0, 0, 0) with all points satisfying \(x^2 + y^2 + z^2 = 9\) lying on the surface. The radius extends 3 units in all directions from the center.

Step 4: Sketching the Sphere

In your sketch, draw three coordinate axes: the x-axis, y-axis, and z-axis, all intersecting at the origin. Since it's a sphere, draw a circle centered at the origin with a radius of 3 units, extending symmetrically in the positive and negative directions on all three axes. This circle represents the intersection of the sphere with the plane, while the full shape is spherical.

Key Concepts

Equation of a Sphere

One of the fundamental elements in 3D geometry is the equation of a sphere. A sphere is a perfectly symmetrical object, where every point on its surface is equidistant from its center. The standard form of a sphere's equation is provided by \[ x^2 + y^2 + z^2 = r^2 \] where \( (x, y, z) \) are the coordinates of any point on the sphere's surface, and \( r \) is the sphere's radius. For instance, the equation \( x^2 + y^2 + z^2 = 9 \) represents a specific sphere. Here, the sphere is centered at the origin, or point \( (0, 0, 0) \), and it encompasses all points satisfying the equation where the sum of the squares of the coordinates equals 9. This means it includes and all points exactly 9 units away from the origin squared.

3D Geometry

Three-dimensional (3D) geometry introduces an extra dimension beyond what's visible in 2D shapes, like squares and circles. The three dimensions are typically represented by the x, y, and z axes. These axes all meet at a point known as the origin.

• The x-axis runs left to right.
• The y-axis runs up and down.
• The z-axis adds depth, running forward and backward.

In the context of spheres, these axes are used to define the position of any point on the sphere. The equation \( x^2 + y^2 + z^2 = 9 \) describes a three-dimensional sphere in this space, with the surface being equidistant from the origin in all directions. This method of representation is critical for visualizing spheres and other 3D shapes.

Radius of a Sphere

The radius of a sphere holds significant implications for its size and structure. Specifically, the radius denotes the distance from the center of the sphere to any point on its surface. \[ r = \sqrt{r^2} \]Using the given equation \( x^2 + y^2 + z^2 = 9 \), we identify the radius as 3 since \( r^2 = 9 \), making \( r = \sqrt{9} = 3 \). This value means every point on the surface of this sphere is 3 units from the center at \((0, 0, 0)\). The radius not only determines the sphere’s size but also affects its graph when drawn. With a radius of 3, the sphere extends symmetrically along the x, y, and z axes, creating a balanced and uniform shape in every direction. Understanding the radius is essential for interpreting and sketching spheres in mathematical problems.