Problem 62 - University Calculus: Early Transcendentals

Question

Find equations for the spheres whose centers and radii are given. $$ (0,-1,5) \quad 2 $$

Step-by-Step Solution

Answer

The equation of the sphere is \(x^2 + (y + 1)^2 + (z - 5)^2 = 4\).

Step 1: Identify the Center and Radius

The center of the sphere is given as \((0, -1, 5)\), and the radius is \(2\).

Step 2: Recall the Standard Equation of a Sphere

The standard equation of a sphere with center \((h, k, l)\) and radius \(r\) is: \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]

Step 3: Substitute the Center Coordinates

Substitute \(h = 0\), \(k = -1\), \(l = 5\) into the equation: \[(x - 0)^2 + (y + 1)^2 + (z - 5)^2 = r^2\]

Step 4: Substitute the Radius

Substitute \(r = 2\) into the equation: \[(x - 0)^2 + (y + 1)^2 + (z - 5)^2 = 2^2\]

Step 5: Simplify the Equation

Simplify the equation by performing the operations: \[x^2 + (y + 1)^2 + (z - 5)^2 = 4\] This is the equation of the sphere.

Key Concepts

Center of a Sphere

The center of a sphere is a critical concept in 3D geometry. It defines the fixed point from which every point on the surface of the sphere is equidistant. This fixed distance is known as the radius. When someone refers to the center of a sphere, they are pointing out a specific point in three-dimensional space. In our exercise, the center is given as

• Coordinates: The center is defined as this case, the point (0, -1, 5). Each coordinate represents the position along the x, y, and z axes in 3D space.
• Importance: The center helps in constructing the sphere's equation and is pivotal in analyzing problems in three dimensions. Understanding the center allows you to visualize the sphere in a complex space.

With these coordinates, you can determine not only the position of the sphere in space but also how it may interact with other geometric objects.

Radius of a Sphere

The radius of a sphere plays a key role in defining the size of the sphere. It is the constant distance from the center of the sphere to any point on its surface. It helps establish how large or small a sphere is.

• Definition: In mathematics, the radius is the line segment from the center of the sphere to a point on its surface. In this exercise, the radius is given as 2.
• Relevance: Knowing the radius allows you to not only identify the size but also aids in calculations like the volume and surface area of the sphere using formulas such as \(V = \frac{4}{3} \pi r^3\) and \(A = 4 \pi r^2\).
• Utility: The radius is crucial in real-world applications, such as determining the size of atoms in chemistry or the dimensions of celestial bodies in astronomy.

Through these insights, the radius not only quantifies the sphere's dimensions but also connects with other geometric and scientific calculations.

Standard Equation of a Sphere

The standard equation of a sphere is the cornerstone for understanding spheres in 3D geometry. It mathematically describes a sphere using algebraic expressions based on its center and radius.

• Equation Form: The formula \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\) expresses a sphere where:
  • \((h, k, l)\) are the coordinates of the center.
  • \(r\) is the radius.
• Customization: You can substitute the values of the center's coordinates and the radius into this equation to tailor it to a specific sphere. For instance, for our sphere, the equation becomes \(x^2 + (y + 1)^2 + (z - 5)^2 = 4\).
• Applications: This equation can solve geometric problems involving spheres, such as finding intersections or calculating distances in 3D spaces.

Understanding this equation aids in visualizing and working with spheres in various mathematical contexts, helping bridge algebra and geometry intuitively.

3D Geometry

3D geometry is an advanced mathematical topic that deals with three-dimensional objects. It extends the study of shapes and figures to a space that has width, depth, and height.

• Fundamentals: Unlike 2D geometry, which deals with flat shapes like circles and squares, 3D geometry includes objects like spheres, cubes, and pyramids. Each of these has volume and surface area.
• Coordinate System: Points in 3D space are represented by three coordinates: x, y, and z, facilitating the study of 3D objects mathematically.
• Relationships: 3D geometry analyzes how different objects, like spheres, interact. It might involve finding distances between objects or determining intersection points.
• Visualization: It aids in developing spatial awareness and visualization skills, essential in fields like engineering and computer graphics.

Grasping 3D geometry concepts broadens the scope of mathematical applications, from simple spheres to complex models used in various scientific and technological fields.