Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the sphere that is tangent to the three coordinate planes if its radius is 6 and its center is in the first octant.

Step-by-Step Solution

Answer

The equation is (x - 6)^2 + (y - 6)^2 + (z - 6)^2 = 36.

Step 1: Understanding the Problem

The problem asks us to find the equation of a sphere that touches all three coordinate planes (x, y, z planes) and has a given radius of 6. The center of the sphere is located in the first octant, meaning coordinates are positive.

Step 2: Determine the Center of the Sphere

Since the sphere is tangent to all three coordinate planes and located in the first octant, its center will have coordinates that are equal to its radius from each plane. Thus, the center of the sphere must be at (6, 6, 6).

Step 3: Equation of the Sphere

The general equation for a sphere with center (h, k, l) and radius r is given by:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]Substituting (h, k, l) = (6, 6, 6) and r = 6, we get:\[(x - 6)^2 + (y - 6)^2 + (z - 6)^2 = 6^2\]

Step 4: Simplify the Equation

Simplifying the above equation:\[(x - 6)^2 + (y - 6)^2 + (z - 6)^2 = 36\]This is the equation of the sphere that satisfies all the given conditions.

Key Concepts

Coordinate Planes

To understand the equation of a sphere, we must first understand the concept of coordinate planes. The coordinate system typically consists of three planes: the x-plane, y-plane, and z-plane. These divide the space into different sections called octants.

- **X-plane**: The plane where the y and z coordinates can take any value, but the x coordinate is zero.
- **Y-plane**: The plane where the x and z coordinates can take any value, but the y coordinate is zero.
- **Z-plane**: The plane where the x and y coordinates can take any value, but the z coordinate is zero.

Coordinate planes are fundamental in determining the position of objects in space. When we say a sphere is tangent to these planes, it means it just touches each plane. This touch occurs at a specific point on the sphere, implying that the distance from the center of the sphere to the plane is equal to the sphere's radius. For a sphere of radius 6, the center must be 6 units away from each plane when it is tangent to all three.

Radius of a Sphere

In geometry, the radius of a sphere is an essential element in understanding its size and position. The radius is the distance from the center of the sphere to any point on its surface. This helps in identifying the size of the sphere and also plays a critical role in its equation.

Consider the sphere in our problem. Its radius is given as 6 units. This is crucial because:

• The radius determines how far the sphere extends from its center in every direction.
• It also helps in connecting the center's location concerning the coordinate planes.

In the formula for the equation of a sphere \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), the radius 'r' is squared and used as a constant that defines the size of the sphere in mathematical terms. For (h, k, l) being (6, 6, 6), the equation becomes the foundation for solving related spatial problems.

First Octant

The concept of the first octant is linked to the positive coordinates in a three-dimensional space. An octant refers to one of the eight sections created when you divide space with the x, y, and z coordinate planes.

The first octant is specifically the section where:

• X-axis values are positive.
• Y-axis values are positive.
• Z-axis values are positive.

This spatial area is where all coordinates are above zero and is generally used when initially finding solutions to three-dimensional problems. For the sphere in this exercise, being in the first octant confirms that its center is made up of positive coordinates (6, 6, 6), ensuring it is effectively and symmetrically positioned in relation to the origin. This understanding simplifies the task of finding the sphere's equation because it restricts the center's possible locations to those also meeting the tangency conditions with the coordinate planes.