Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Find the standard equation of the sphere. Center: \((1,1,5) ;\) radius: 3

Step-by-Step Solution

Answer

The standard equation of the sphere is \((x-1)^2 + (y-1)^2 + (z-5)^2 = 9\).

Step 1: Identify the components of the equation

The standard equation of a sphere is \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\). Here we have to replace \(a, b, c\) with the coordinates of the center and \(r\) with the given radius. Here, the center of the sphere is \((1,1,5)\) and the radius is 3.

Step 2: Substitute the values into the equation

Replace \(a, b, c\) with 1, 1 and 5 respectively, and replace \(r\) with 3 in the standard equation of a sphere. The equation becomes \((x-1)^2 + (y-1)^2 + (z-5)^2 = 3^2\).

Step 3: Simplify the equation

Simplify the equation by squaring the radius. The final equation of the sphere is \((x-1)^2 + (y-1)^2 + (z-5)^2 = 9\).

Key Concepts

Understanding Three-Dimensional Geometry

Three-dimensional geometry allows us to understand the positions, shapes, and sizes of objects in space. Unlike two-dimensional geometry, which deals with flat shapes like circles and squares, three-dimensional geometry includes objects with depth, like spheres, cubes, and cylinders. This branch of mathematics examines the relations between points, lines, and surfaces in three-dimensional space.

It's essential to grasp three-dimensional geometry as it helps us solve problems related to volume, surface area, and spatial relationships. To navigate through three-dimensional geometry, we need a coordinate system, typically utilizing the x, y, and z axes. Each point in this space is defined by a set of three numbers, known as coordinates, written as \(\(x, y, z\)\). Understanding these coordinates allows us to accurately describe the locations and dimensions of shapes like spheres.

Moreover, by leveraging three-dimensional geometry, we can create models, analyze spatial distributions, and visualize concepts that are crucial in fields such as physics, engineering, and computer graphics.

The Standard Form of a Sphere

The standard form of a sphere's equation is a fundamental concept in geometry that assists with identifying and working with spheres in three-dimensional space. This form is expressed as \(\((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\)\). Each part of this equation provides vital information about the sphere's properties.

In the equation:

• \(a, b, c\) are the coordinates of the sphere's center. They define the central point around which the sphere is perfectly symmetrical.
• \(r\) is the radius, determining the size of the sphere. It's the distance from the center to any point on the surface.

The standard form allows us to quickly understand the sphere's position and size in space just by analyzing these components. It's crucial when working with spheres to substitute the given center and radius into this form to describe the sphere's equation accurately.

This form not only simplifies calculations but also provides a clear way to visualize the sphere mathematically, forming the basis for more complex spatial reasoning and calculations.

The Center and Radius of a Sphere

A sphere's center and radius are two critical elements defining its shape and position in space. The center is a point inside the sphere. It is equidistant from all points on the surface. In a three-dimensional space, it has the coordinates \(\((a, b, c)\)\), acting like the sphere's anchor, balancing its properties symmetrically around this point.

The radius, denoted by \(r\), extends from the center to any point on the surface. It effectively measures the sphere's size, with a longer radius resulting in a larger sphere and vice versa. The radius plays a vital role in calculations involving the sphere, such as determining volume or surface area.

To find the standard equation of a sphere, we need the center and radius data. Inserting these values into the standard sphere equation allows you to fully describe the sphere in a mathematical sense. Whether in academic studies or real-world applications like computer simulations or physics problems, understanding these concepts is key to harnessing the full potential of working with spheres.