Problem 49 - Precalculus with Calculus Previews

Question

Find an equation of a sphere that satisfies the given conditions. $$ \text { Center }(1,1,4) ; \text { tangent to the } x y \text { -plane } $$

Step-by-Step Solution

Answer

The equation of the sphere is \((x-1)^2 + (y-1)^2 + (z-4)^2 = 16\).

Step 1: Understanding the Problem

We need to find the equation of a sphere. A sphere is defined by its center and radius. We are given the center of the sphere as (1, 1, 4) and told that the sphere is tangent to the xy-plane.

Step 2: Sphere Equation Formula

The general equation for a sphere with center ext (h, k, l) and radius is: . Plug in the center (1, 1, 4) into this equation. The sphere equation becomes: .

Step 3: Determine Sphere's Radius

Since the sphere is tangent to the xy-plane, the distance from the center of the sphere to the xy-plane is equal to the radius of the sphere. The perpendicular distance from the center (1, 1, 4) to the xy-plane is the z-coordinate of the center, which is . Therefore, the radius of the sphere is 4.

Step 4: Forming the Equation

Substitute the center (1, 1, 4) and the radius 4 into the sphere equation to form the specific equation: .

Step 5: Final Sphere Equation

The equation of the sphere with center (1, 1, 4) and radius 4 is . This is the equation that defines the sphere based on the given conditions.

Key Concepts

Center of the Sphere

The center of a sphere is a crucial component when defining its equation. In this specific exercise, we are given the center at coordinates \((1, 1, 4)\). This means that the sphere is positioned in three-dimensional space with its center at the point where:

• The x-coordinate is 1
• The y-coordinate is 1
• The z-coordinate is 4

The center's coordinates directly influence the structure and position of the sphere's equation.The equation of a sphere is generally represented as: \[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]where - \(h, k, l\) are the coordinates of the center, and - \(r\) is the radius. Substituting in the center coordinates from our given problem (1, 1, 4), we start forming the sphere's equation.

Radius of the Sphere

The radius of a sphere represents the distance from the center to any point on its surface. In our exercise, the sphere is tangent to the \(xy\)-plane.This tangency condition provides vital information about the sphere's radius.Tangent to the \(xy\)-plane means the shortest distance between the center of the sphere and the \(xy\)-plane is equal to the radius.In this case, the z-coordinate of the center determines this distance.Thus, for our sphere with the center at \((1, 1, 4)\), the radius \(r\) is the z-coordinate value, i.e., \(r = 4\). Hence, we already know the formula for calculating the sphere's radius based on the height above the \(xy\)-plane. Now, integrate this radius back into the sphere's equation, which helps in finalizing its definition.

Tangent to the Plane

When a sphere is tangent to a particular plane, it touches the plane at precisely one point. For this exercise, the sphere is tangent to the \(xy\)-plane.This condition simplifies our calculations considerably.

• The tangency means that the perpendicular distance between the sphere's center and the plane is exactly equal to the sphere's radius.
• This relationship is especially significant as it allows us to set the radius as the z-coordinate from the center \((1, 1, 4)\), which is \(4\).

Using this information, we can ascertain the complete sphere equation:\[(x - 1)^2 + (y - 1)^2 + (z - 4)^2 = 4^2\]This tangency ensures that only the specified radius fits perfectly in contact with the plane, comprehensively determining the sphere's geometry.