Problem 92 - Precalculus with Calculus Previews

Question

Describe geometrically all points in 3-space whose coordinates satisfy the given condition(s). $$ x^{2}+y^{2}+(z-1)^{2}=4, z=2 $$

Step-by-Step Solution

Answer

A circle with radius \( \sqrt{3} \) centered at \((0,0,2)\).

Step 1: Understand the First Equation

Consider the equation \( x^2 + y^2 + (z-1)^2 = 4 \). This is the equation of a sphere in three-dimensional space, centered at the point \((0, 0, 1)\) with a radius of 2.

Step 2: Analyze the Second Condition

The condition \( z = 2 \) specifies a horizontal plane in three-dimensional space at the height where the z-coordinate is constantly 2.

Step 3: Intersection of the Sphere and the Plane

To find the geometrical description of the points, consider where the plane \( z = 2 \) intersects with the sphere. Substitute \( z = 2 \) into the sphere's equation: \( x^2 + y^2 + (2-1)^2 = 4 \).

Step 4: Simplify the Equation

Substituting \( z = 2 \) into the sphere equation transforms it into \( x^2 + y^2 + 1 = 4 \), which simplifies to \( x^2 + y^2 = 3 \).

Step 5: Describe the Geometrical Shape

The equation \( x^2 + y^2 = 3 \) describes a circle in the xy-plane, centered at \((0, 0)\) with a radius of \( \sqrt{3} \), located at the height \( z = 2 \).

Key Concepts

Sphere

A sphere is a perfectly symmetrical 3D shape where every point on its surface is an equal distance from its center. This distance is known as the radius.

• The equation for a sphere is typically: \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \). Here, \(h, k, l\) are the coordinates of the center, and \(r\) is the radius.
• In the given exercise, the sphere is represented by \( x^2 + y^2 + (z - 1)^2 = 4 \). This tells us that the sphere's center is at \(0, 0, 1\) and its radius is 2.

Understanding the components of a sphere's equation helps us easily locate its position and size in space. Spheres are basic elements in geometry, important because they help in understanding complex interactions in 3D spaces.

Plane

A plane extends infinitely in two dimensions and is a flat, flat surface without edges. It's defined mathematically by a linear equation.

• The simplest plane, parallel to one of the coordinate planes, can take the form such as \( z = c \), where all points on the plane share the same z-coordinate.
• In the context of the exercise, the plane is described by \( z = 2 \), indicating a horizontal plane parallel to the xy-plane, located two units above the origin along the z-axis.

Recognizing planes is essential to understanding complex geometric configurations. They can intersect other shapes in interesting ways, forming various geometric features.

Intersection of Geometric Shapes

The intersection of two geometric shapes occurs where their equations simultaneously hold, producing common points or curves.

• When intersecting a plane and a sphere, you analyze how the flat infinite plane intersects the curved surface of the sphere.
• For the sphere \( x^2 + y^2 + (z-1)^2 = 4 \) and the plane \( z = 2 \), substituting \( z = 2 \) into the sphere's equation simplifies to \( x^2 + y^2 = 3 \), indicating a circular intersection.

Taking the intersection between these shapes allows us to visualize where the plane "cuts through" the sphere, forming a smaller, two-dimensional geometric shape—often a circle.

Circle in Space

A circle in space represents a set of points equidistant from a central point within a plane.

• It can appear as the cross-section of a more complex shape, like when a plane cuts through a sphere.
• In the example given, the intersection formed where \( x^2 + y^2 = 3 \) describes a circle in the plane \( z = 2 \).
• This circle has its center at \(0, 0, 2\) and radius \( \sqrt{3} \), derived from simplifying the intersected sphere equation.

Recognizing and analyzing circles in 3D geometry can provide insights into spatial relationships and dimensions across different mathematical and physical contexts.