Problem 11 - Thomas Calculus

Question

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}+(z+3)^{2}=25, \quad z=0$$

Step-by-Step Solution

Answer

The intersection is a circle with radius 4, centered at (0,0) in the XY-plane.

Step 1: Understand the Equation

The first equation is \( x^{2}+y^{2}+(z+3)^{2}=25 \). This represents a sphere centered at the point \((0, 0, -3)\) with a radius of 5 units in 3D space.

Step 2: Analyze the Constraint

The second equation \( z=0 \) is a constraint that restricts the points to the plane \( z=0 \), which is the XY-plane in 3D space.

Step 3: Find the Intersection

To find the geometry of the intersection, substitute \( z=0 \) into the sphere's equation: \( x^2 + y^2 + (0+3)^2 = 25 \). Simplifying, we get \( x^2 + y^2 + 9 = 25 \).

Step 4: Simplify and Interpret

Simplify the equation from the previous step: \( x^2 + y^2 = 16 \). This represents a circle centered at the origin \((0,0)\) with a radius of 4 in the XY-plane.

Key Concepts

Understanding 3D Space

In mathematics, when we refer to "3D space," we are discussing a space that has three dimensions: length, width, and height. These dimensions are often described using the axes in a Cartesian coordinate system, namely the x-axis, y-axis, and z-axis.

The ability to visualize 3D space extends our understanding beyond the flat dimensions of 2D space, which only includes length and width (or height).

• The x-axis usually runs horizontally.
• The y-axis runs perpendicular to the x-axis, sometimes also horizontally, dependent on orientation.
• The z-axis typically represents height and runs perpendicular to both the x and y axes.

This three-dimensional space allows us to model objects as we see in the real world, providing depth and a full scope of geometrical representation.

Exploring Spheres

A sphere in 3D geometry is a perfectly symmetrical object. Every point on its surface is equidistant from a fixed point known as the center.

The formula for a sphere in a coordinate system is \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\). Here, \((a, b, c)\) represents the center coordinates of the sphere, and \(r\) is the radius.

• If we examine the equation given in the exercise, \(x^{2}+y^{2}+(z+3)^{2}=25\), we can identify the center as \((0, 0, -3)\) and the radius as \(5\) (since \(r^2 = 25\)).
• To analyze how a sphere interacts with other geometric entities like planes, we can substitute and solve equations to find intersections.

Understanding spheres in this context allows us to interpret complex geometrical problems and visualize intersections, such as circles, which are derived when spheres intersect planes.

Coordinate Systems for Geometry

Coordinate systems are fundamental in representing and solving geometric problems, especially in analytic geometry. They give a way to describe the location of points and geometrical shapes in space.

The Cartesian coordinate system, named after René Descartes, is the most conventional system. It uses ordered pairs or triplets of numbers to determine the position of a point in space.

• A point in two dimensions is described with an ordered pair \((x, y)\), where \(x\) and \(y\) are the coordinates along the x-axis and y-axis respectively.
• In three dimensions, a point expands to \((x, y, z)\), adding the z-coordinate to allow for elevation or depth.

In the given problem, this system is utilized to express the location of a sphere and how it intersects the xy-plane when \(z = 0\). This highlights the power of coordinate systems to describe complex geometric scenarios with simple mathematical expressions.