Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the sphere whose center is \((2,4,5)\) and that is tangent to the \(x y\) -plane.

Step-by-Step Solution

Answer

The sphere's equation is \((x - 2)^2 + (y - 4)^2 + (z - 5)^2 = 25\).

Step 1: Understand the Tangent Condition

The sphere being tangent to the \(xy\)-plane means that the distance from the center of the sphere \((2,4,5)\) to the \(xy\)-plane is equal to the radius. The \(xy\)-plane can be described by the equation \(z = 0\). Therefore, the radius is simply the \(z\)-coordinate of the center, which is 5.

Step 2: Apply the Sphere Equation Formula

The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is: \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). Substituting the given center \((2,4,5)\) and radius \(5\), the equation becomes: \((x - 2)^2 + (y - 4)^2 + (z - 5)^2 = 5^2\).

Step 3: Simplify the Equation

Simplify the equation obtained in the previous step: \((x - 2)^2 + (y - 4)^2 + (z - 5)^2 = 25\). This is the final equation of the sphere.

Key Concepts

Tangent Condition

When discussing spheres, the tangent condition refers to a situation where a sphere just touches a plane. This point of contact is called tangency. In this exercise, the sphere is tangent to the \(xy\)-plane. The \(xy\)-plane is represented by the equation \(z = 0\).

• The sphere's center is at \((2,4,5)\).
• Tangency means the distance from the center of the sphere to the plane is exactly the radius.

For this problem, the sphere is tangent to the \(xy\)-plane. The center's \(z\)-coordinate of 5 is crucial. This value tells us how far up the sphere is positioned from the plane, establishing it as the radius of the sphere. Understanding tangency is key in positioning spheres in 3D space.

Sphere Center

The center of a sphere is the fixed point from which every point on the sphere's surface is equidistant. It's a crucial part of the sphere's equation. In our problem, the center is at \((2, 4, 5)\). This location includes specific coordinates:

• \(x\)-coordinate: 2
• \(y\)-coordinate: 4
• \(z\)-coordinate: 5

Knowing the center helps in formulating the equation of the sphere. It shows the point in 3D space where the middle of the sphere is situated. You can think of the center as the origin of the sphere in its own coordinate system. Every formula involving spheres anchors itself around this core point.

Radius

The radius is the distance from the center of the sphere to any point on its surface. It's a key element to describe the size of the sphere. In our problem, because the sphere is tangent to the \(xy\)-plane, the radius is precisely the \(z\)-coordinate of the center which is 5.
Here’s why:

• The distance from the center at \((2, 4, 5)\) to the \(xy\)-plane \(z = 0\) is 5.
• That same distance acts as the radius due to the tangent condition.

The radius is essential in forming the sphere's equation as it defines the sphere's perimeter in 3D space. Knowing the radius helps plot an accurate sphere surrounding its center.

3D Coordinates

3D coordinates consist of three values \((x, y, z)\), representing a point in three-dimensional space. For a sphere, these coordinates explain location and orientation:

• The \(x\)-coordinate shows left-right positioning.
• The \(y\)-coordinate explains forward-backward movement.
• The \(z\)-coordinate describes up-down altitude.

These coordinates are vital for identifying the position of the sphere's center, which in this problem is \((2, 4, 5)\). Understanding how to work with 3D coordinates is fundamental for solving problems that involve spatial objects. When calculating with spheres, combining all three dimensions gives a complete overview of a sphere’s position and enables accurate measurements of distances, like the radius.