Problem 53 - Thomas Calculus

Question

Find equations for the spheres whose centers and radii are given in Exercises \(51-54 .\) $$\frac{\text { Center }}{\left(-1, \frac{1}{2},-\frac{2}{3}\right)} \frac{\text { Radius }}{\frac{4}{9}}$$

Step-by-Step Solution

Answer

The equation is \((x + 1)^2 + \left(y - \frac{1}{2}\right)^2 + \left(z + \frac{2}{3}\right)^2 = \frac{16}{81}\)."

Step 1: Identify the Center and Radius of the Sphere

The exercise gives us the center of the sphere as \((-1, \frac{1}{2}, -\frac{2}{3})\) and the radius as \(\frac{4}{9}\).

Step 2: Standard Equation of a Sphere

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

Step 3: Substitute the Given Values

Substitute the center \((-1, \frac{1}{2}, -\frac{2}{3})\) and radius \(\frac{4}{9}\) into the standard equation: \[(x + 1)^2 + \left(y - \frac{1}{2}\right)^2 + \left(z + \frac{2}{3}\right)^2 = \left(\frac{4}{9}\right)^2\]

Step 4: Simplify the Equation

Calculate the square of the radius \(\frac{4}{9}\): \[\left(\frac{4}{9}\right)^2 = \frac{16}{81}\]Thus, the final equation of the sphere will be: \[(x + 1)^2 + \left(y - \frac{1}{2}\right)^2 + \left(z + \frac{2}{3}\right)^2 = \frac{16}{81}\]

Key Concepts

Center of a Sphere

The center of a sphere is a crucial component in 3D coordinate geometry. It serves as the point from which every point on the sphere's surface is equidistant. In the exercise provided, the center of the sphere is given as \((-1, \frac{1}{2}, -\frac{2}{3})\). This set of coordinates represents the exact point in a three-dimensional space that is at the core of the sphere. We can think of this point as the sphere's 'home base'.

This center point is indicated by three coordinates \(h, k, l\), which define its position along the x, y, and z axes, respectively. Each axis corresponds to a dimension in space, helping to localize the position of the center within three-dimensional space:

• \(h\) is the x-coordinate, which in this case is -1.
• \(k\) is the y-coordinate, here given as \(+\frac{1}{2}\).
• \(l\) is the z-coordinate, given as \(-\frac{2}{3}\).

To identify the center in a real-world context, imagine a point hovering somewhere, determined by these coordinates, defining where the sphere's geometry begins.

Radius of a Sphere

The radius is another fundamental element in defining a sphere. It represents the distance from the center to any point on the surface of the sphere. In this exercise, the radius is given as \(\frac{4}{9}\). This means that every point along the surface of the sphere is exactly \(\frac{4}{9}\) units away from the center coordinate \((-1, \frac{1}{2}, -\frac{2}{3})\).

The significance of the radius lies in its role in determining the size of the sphere. Larger radii mean larger spheres and vice versa. Hence, it has a profound influence on the spatial extent of the sphere. Mathematically, the radius \(r\) appears in the equation of a sphere as \(r^2\), representing its squared value:

For example, the equation used was \[(x + 1)^2 + \left(y - \frac{1}{2}\right)^2 + \left(z + \frac{2}{3}\right)^2 = \frac{16}{81}\] where the right side \(\frac{16}{81}\) is \(\left(\frac{4}{9}\right)^2\).

Understanding how to determine and use the radius directly impacts how we visualize shapes in 3D space.

3D Coordinate Geometry

3D coordinate geometry is the study of points, lines, and shapes in a three-dimensional space. It expands on the familiar concepts of 2D coordinate geometry by adding a third z-dimension, allowing for a more comprehensive representation of objects like spheres.

In this context, the coordinates \(x, y, z\) define positions within this 3D space. Each point is a precise location that can be occupied by any geometric entity. When we talk about spheres, we use these coordinates to describe both the center and points on the surface.

• **Center**: Described by \(h, k, l\), it is the anchor point of the sphere.
• **Equation of Sphere**: The equation \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\) leverages these coordinates to define every point on the sphere's surface.

The inclusion of all three axes allows us to work with real-world objects accurately by considering their depth, height, and width. This geometry is crucial in fields like computer graphics, engineering, and physics, where modeling real-world objects with accuracy is essential. It helps in understanding spatial relationships and measurements within three-dimensional contexts.