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Chapter 11: Problem 39

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.

Short Answer

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

Step by step solution

01

Understand the Problem

To find the equation of a sphere tangent to the three coordinate planes, we need to determine the center and radius of the sphere. The sphere is in the first octant with a radius of 6.
02

Use Sphere Equation Formula

The general equation for a sphere with center \((h, k, l)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). Here, \(r = 6\).
03

Determine Center Coordinates

Since the sphere is tangent to the coordinate planes, each coordinate of the center \((h, k, l)\) equals the radius. Therefore, \(h = 6\), \(k = 6\), and \(l = 6\).
04

Formulate the Equation

Substitute \(h = 6\), \(k = 6\), \(l = 6\), and \(r = 6\) into the sphere equation formula: \[(x - 6)^2 + (y - 6)^2 + (z - 6)^2 = 36.\]
05

Write the Final Equation

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Coordinate Planes
Coordinate planes form the foundation for organizing objects and points in a three-dimensional space. These planes are essentially flat surfaces that divide space into different sections. In 3D space, there are three primary coordinate planes:
  • The **xy-plane**, which extends horizontally and vertically, leaving out the z-axis.
  • The **xz-plane**, which extends horizontally and along the front-to-back axis, excluding the y-axis.
  • The **yz-plane**, which extends vertically and along the front-to-back axis, excluding the x-axis.
Each plane helps in determining a location or position by using pairs of numbers, which correspond to where a point lies on these intersecting planes. When we talk about the equation of a sphere tangent to these planes, it means the sphere lightly touches each of these planes.
Exploring the First Octant
The three coordinate planes divide space into eight sections, known as octants. The **first octant** is where all three coordinates, x, y, and z, are positive. This section is intuitively easier to understand and is often the first area studied in beginning courses on three-dimensional geometry, hence the name 'first octant.'

When a sphere's center is in the first octant, all coordinates of the center are positive values. For a sphere tangent to all three coordinate planes, like in our problem, this placement in the first octant means the sphere just touches each plane at exactly one point, ensuring each coordinate is equal to the radius.
Understanding Tangency
"Tangency" is the term used when a curve or surface touches another object at a single point. When a sphere is tangent to a plane, like each coordinate plane in our exercise, it means the sphere just meets each plane without cutting through.

For our specific sphere, it is tangent to the three coordinate planes, implying that the center of the sphere is also the same distance, equal to the radius, from each of these planes. This particular characteristic is very important in determining the coordinates of the center of the sphere.
Deriving the Equation of a Sphere
The equation of a sphere is a simple yet profound expression in geometry that helps us understand the spatial boundaries of a sphere based on its center and radius. The general formula for the equation of a sphere is:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]Where:
  • \(h, k, l\) are the coordinates of the center of the sphere.
  • \(r\) is the radius of the sphere.
Using this formula, you can calculate and define any sphere given its center and radius. In our problem, since the sphere is tangent to each coordinate plane and located in the first octant, the center coordinates \((h, k, l)\) are all equal to the radius \(r\). Plugging in these values gives us a concrete mathematical representation of the sphere's surface in the 3D space.

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