Problem 47 - Calculus an Applied Approach

Question

In Exercises \(47-50,\) sketch the \(x y\) -trace of the sphere. $$ (x-1)^{2}+(y-3)^{2}+(z-2)^{2}=25 $$

Step-by-Step Solution

Answer

The \(xy\)-trace of the sphere \((x-1)^{2}+(y-3)^{2}+(z-2)^{2}=25\) is a circle centered at \((1,3)\) on the \(xy\) plane with a radius \(\sqrt{21}\).

Step 1: Identify the Center and Radius of the Sphere

The general formula of a sphere in a 3D coordinate system is \((x-h)^{2}+(y-k)^{2}+(z-l)^{2}=r^2\), where \((h,k,l)\) is the center of the sphere and \(r\) is the radius. Comparing this with given equation, \((x-1)^{2}+(y-3)^{2}+(z-2)^{2}=25\), implies the center is at point \((1,3,2)\) and the radius \(r\) is \(5\) (given 25 is the square of 5).

Step 2: Substitute Z=0

To find the \(xy\)-trace of the sphere, we'll find its intersection with the \(xy\) plane, which corresponds to \(z=0\). Substituting \(z=0\) into the equation of the sphere, we have \((x-1)^{2}+(y-3)^{2}+(0-2)^{2}=25\). This reduces to \((x-1)^{2}+(y-3)^{2}=21\), which is the equation of a circle on the \(xy\) plane with center \((1,3)\) and radius \(\sqrt{21}\).

Step 3: Sketching the Result

This circle is the \(xy\)-trace of the sphere. It's centered at the point \((1,3)\) with a radius \(\sqrt{21}\). To sketch this circle, identify the point \((1,3)\) on the \(xy\) plane and then draw a circle around that point, such that every point on the circle is \(\sqrt{21}\) units away from the center.

Key Concepts

3D Coordinate System

The 3D coordinate system is an extension of the two-dimensional Cartesian system, commonly used to describe the position of points in space. Unlike the 2D plane you may be familiar with, which only requires two coordinates (the x and y axes), the 3D system includes an additional dimension, the z-axis, which measures depth. Imagine the axes as intersecting lines: the x-axis runs horizontally, the y-axis vertically, and the z-axis perpendicular to both, creating a three-dimensional grid. Each point in this space can be identified by a unique set of three numbers, \(x, y, z\), representing its coordinates along these axes.

When working in three dimensions, objects like spheres, cylinders, and other solids can be analyzed with respect to this grid. This allows for complex shapes to be studied in terms of their relations with the coordinate planes and other geometric features.

Equation of a Sphere

The sphere is one of the most fundamental shapes in geometry, and its equation in a 3D coordinate system provides a way to describe its entire surface. The standard form of the equation is \( (x-h)^{2} + (y-k)^{2} + (z-l)^{2} = r^{2} \), where \(h, k, l\) are the coordinates of the sphere's center, and \(r\) is its radius. Remember, all points that satisfy this equation are a fixed distance \(r\) from the center, thus forming the sphere.

Understanding this equation is crucial, as it is the basis for analyzing characteristics like volume, surface area, and in this specific exercise, its \(xy\)-trace. The center and radius extracted from this equation define the sphere's size and position in space and can be related back to the 3D coordinate system for a visual understanding of its orientation and dimensions.

Intersection with the xy Plane

Intersecting a 3D object with one of the coordinate planes involves finding all the points that lie both on the object and on the plane. For the \(xy\)-plane, this means setting the \(z\)-coordinate to zero, since the \(xy\)-plane is defined by \(z = 0\). In practice, you substitute \(z = 0\) into your 3D shape's equation, simplifying it into a 2D equation.

For spheres, this operation usually results in the equation of a circle, since the intersection of a sphere and a plane is typically circular. Solving for the \(xy\)-trace provides the perspective of looking directly down onto the shape, observing the circular cross-section, which is represented by its equation in the \(xy\)-plane. This intersection reveals valuable information about the spatial relations and dimensions of the 3D object in reference to the specified plane.

Sketching Circles in Coordinate Geometry

Drawing circles in coordinate geometry involves first identifying the center of the circle and then marking points that are at a distance equal to the radius away from the center. With the equation in the form \( (x-h)^{2} + (y-k)^{2} = r^{2} \) where \( (h, k) \) is the center and \(r\) is the radius, you can plot the center directly onto the coordinate grid.

From there, you use the radius to determine points on the perimeter, known as the circumference, by moving \(r\) units in all directions––up, down, left, and right––from the center. Connecting these points with a smooth, consistent curve will result in a circle. When sketching, it's helpful to also mark points that are at \(45^\text{o}\) angles from the center since they help make the circle evenly rounded. This technique allows for a precise circle to be represented on a grid, aligning with the foundational aspects of coordinate geometry.