Problem 41 - Precalculus with Calculus Previews

Question

Sketch the graph of the given equation. $$ (x-1)^{2}+(y-1)^{2}+(z-1)^{2}=1 $$

Step-by-Step Solution

Answer

The graph is a sphere centered at (1, 1, 1) with a radius of 1.

Step 1: Identify the type of graph

The given equation \((x-1)^{2}+(y-1)^{2}+(z-1)^{2}=1\) is a standard form of the equation for a sphere. All terms are squares, and they add up to 1, which is the equation of a sphere centered at some point with radius 1.

Step 2: Determine the center and radius

Compare the equation to the general form of a sphere's equation \((x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}\). Here, \(a=1, b=1, c=1\), and \(r=1\). Thus, the sphere is centered at the point \((1, 1, 1)\) and has a radius of 1.

Step 3: Sketch the sphere

To sketch the sphere, start by plotting the center \((1, 1, 1)\) on a 3D coordinate system. Since the radius is 1, the sphere extends 1 unit away from the center in all directions within the 3D space. Use symmetry and the radius limit to guide where the surface of the sphere should be relative to the center.

Key Concepts

Sphere Equation

A sphere equation in 3D geometry is a mathematical representation that describes the set of all points in space equidistant from a fixed point, called the center. The general form of a sphere equation is given by:\[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\]where

• \( (a, b, c) \) denotes the sphere’s center,
• \( r \) is the radius— the fixed distance from the center to any point on the sphere's surface.

In the sphere equation \[(x-1)^{2}+(y-1)^{2}+(z-1)^{2}=1\]we see that the sum involves squared differences for three coordinates (x, y, z) which means the equation is in standard form. This form makes it simple to identify the center and radius of the sphere easily from the equation components.

Center of Sphere

The center of a sphere is the point from which every point on the surface of the sphere is equidistant. This point is crucial in determining the sphere’s position in 3D space. By comparing the given equation:\[(x-1)^2 + (y-1)^2 + (z-1)^2 = 1 \]with the standard sphere equation:\[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\]we can identify the values of \(a\), \(b\), and \(c\) which give us the coordinates of the sphere's center. For the equation above, the center is:

• \( a = 1 \)
• \( b = 1 \)
• \( c = 1 \)

Hence, the center of the sphere is the point \((1, 1, 1)\).Looking at the center helps in sketching and visualizing the sphere properly, especially when plotting it on a coordinate system.

Radius of Sphere

The radius of a sphere is the constant distance from its center to any point on its surface. It defines the size of the sphere and influences how far the surface extends in all directions from the center in 3D space. In the general sphere equation \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\), the radius is the square root of the number on the right side of the equation. For instance, in the equation:\[(x-1)^2 + (y-1)^2 + (z-1)^2 = 1\]the value on the right is 1, so the radius \(r\) is:\[r = \sqrt{1} = 1 \]This tells us that the sphere extends 1 unit away from the center in all directions. Recognizing the radius allows you to accurately measure and sketch the sphere’s dimensions and ensures that you correctly scale its placement relative to the coordinate system's axes.