Problem 8 - Calculus: Early Transcendentals

Question

Find an equation of the sphere with center at (1,1,1) and radius 2 .

Step-by-Step Solution

Answer

The equation of the sphere is \\( (x - 1)^2 + (y - 1)^2 + (z - 1)^2 = 4 \\).

Step 1: Identify Sphere Formula

The general equation of a sphere with center \(h, k, l\) and radius \ r \ is \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \). Identify the values of \(h, k, l, \text{and} \ r \) for the problem.

Step 2: Substitute Center Coordinates

Insert the center coordinates \(h = 1, k = 1, l = 1\) into the formula: \( (x - 1)^2 + (y - 1)^2 + (z - 1)^2 = r^2 \).

Step 3: Plug in the Radius

Replace \(r\) with \(2\) in the equation: \((x - 1)^2 + (y - 1)^2 + (z - 1)^2 = 2^2\). This set the square of the radius in the equation.

Step 4: Simplify the Equation

Calculate \(r^2 = 2^2 = 4\) and substitute back: \((x - 1)^2 + (y - 1)^2 + (z - 1)^2 = 4\). This is the final equation of the sphere.

Key Concepts

Center of a Sphere

The center of a sphere is a point in three-dimensional space that is equidistant from all points on the surface of the sphere. Imagine a perfect ball within three dimensions, and the center is, as you might expect, right in the middle.
In our exercise, the sphere's center is at the point \(1, 1, 1\).

• The coordinates given are \(h, k, l\), representing the position of the center in a 3D space.
• These coordinates tell us exactly where the middle of our sphere is located.

Understanding this point is essential because it is used in the equation of the sphere. You'll notice the equation \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \) has \( h, k, l \) directly linked to the center of the sphere. Knowing the center helps in creating the equation needed to understand everything about the sphere, including how it fits and interacts within its 3D space.

Radius of a Sphere

The radius of a sphere is the fixed distance from its center to any point on its surface. Think about how you stretch a string from the center of a rubber ball to its edge; that length is the radius.
This specific measurement is crucial because it directly affects the sphere's size and shape in space. In our given problem, the radius is 2.

• The equation \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \) uses the radius squared \( r^2 \)
• So, with a radius of 2, you'll calculate \( r^2 = 2^2 = 4 \).

This number, 4, is used in the equation to define the space the sphere occupies. The radius being squared means even a small change in the radius will significantly affect the sphere's volume and boundary, a concept highly relevant in mathematical and real-world applications.

3D Geometry

Three-dimensional geometry is all about shapes and figures that have depth, width, and height. In the realm of 3D geometry, spheres are a perfect representation of symmetry and balance. They create a seamless boundary all around their center.
Understanding 3D geometry includes being able to visualize these shapes within a space that extends in three directions. For our sphere, we work in a space with coordinates \(x, y, z\), giving us a full picture of where the sphere 'lives' and 'breathes.'

• A sphere is defined not just by its center and radius but by how it interacts with the surrounding 3D space.
• It is a fundamental part of studying how shapes behave beyond the plane.

Equations like \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \) allow us to explore how these figures relate to each other, paving the way for more advanced studies in fields like physics, engineering, and computer graphics.