Problem 51 - Calculus an Applied Approach

Question

In Exercises \(51-54,\) sketch the \(y z\) -trace of the sphere. $$ x^{2}+(y+3)^{2}+z^{2}=25 $$

Step-by-Step Solution

Answer

The \(yz\)-trace of the sphere can be represented as a circle in the \(yz\)-plane. It is centered at (0,-3) and has a radius of 5 units.

Step 1: Identify the sphere equation

The equation \(x^{2}+(y+3)^{2}+z^{2}=25\) can be identified as the equation of a sphere. In this equation, the center of the sphere \((h, k, l)\) is at (0, -3, 0), and the radius r equals to \(\sqrt{25}=5\)

Step 2: Identify intercepts

To sketch the sphere, we need to find the intercept points along the \(y\) and \(z\)-axis. When \(x=0\), we have the trace equation \[(y+3)^{2}+z^{2}=25\]. The intercept values occur when either \(y+3=0\) or \(z=0\) since in both cases we obtain a circle equation of radius 5 centered on the y-axis at -3 (for the z=0) or centered on the z-axis at -3 (for y+3=0). These constitute a cross section of the sphere in the \(yz\)-plane. The intercepts based on the circle equation are at \[y = -3 \pm 5\] and \[z = -3 \pm 5\], which gives 4 intersection points to be (-8, 0), (2, 0), (0, -8) and (0, 2)

Step 3: Sketch the \(yz\)-plane

Plot these points on the \(yz\)-plane. Draw a circle that goes through all the four points. This represents the cross section of the sphere as it intersects with the \(yz\)-plane, which is the required \(yz\)-trace of the given sphere. Make sure that the center of circle lies at (0,-3) and its radius is 5 units.

Key Concepts

Sphere Geometry

Sphere geometry is a topic in mathematics that deals with objects having every point on their surface equidistant from their center. This distance is known as the radius.

In mathematical terms, a sphere's equation can be represented as \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \), where \( (h, k, l) \) represents the center of the sphere in a 3D coordinate system, and \( r \) is the radius. For the given exercise \( x^2 + (y+3)^2 + z^2 = 25 \), by comparing it with the standard form, it is clear that the sphere's center is at \( (0, -3, 0) \) and has a radius of \( 5 \).

Understanding sphere geometry is crucial for visualizing objects in 3D and solving problems related to volume, surface area, and intersections with other shapes.

YZ-Trace

The yz-trace of a sphere provides a 2D cross-sectional view of the sphere when sliced along a plane parallel to the yz-plane, in this case, where \( x = 0 \).

For the given sphere's equation \( x^2 + (y+3)^2 + z^2 = 25 \), by setting \( x = 0 \) we get the yz-trace equation: \[ (y+3)^2 + z^2 = 25 \]. This equation resembles the circle equation in two dimensions, indicating that the yz-trace of the sphere is a circle with center \( (0, -3) \) and radius \( 5 \) units. To sketch this, one would plot the circle on the yz-plane. This 2D representation is helpful for understanding the shape and position of the sphere in a 3D coordinate system.

Intercepts of a Sphere

Intercepts of a sphere refer to the points where the sphere intersects the coordinate axes. In the 3D coordinate system, a sphere can intersect the x, y, and z-axes.

In our specific case \( x^2 + (y+3)^2 + z^2 = 25 \), the intercepts on the y-axis occur when \( x = 0 \) and \( z = 0 \) which gives us \( y = -3 \pm 5 \) resulting in the points \( (0, -8, 0) \) and \( (0, 2, 0) \). Similarly, for the z-axis intercepts, we set \( x = 0 \) and \( y + 3 = 0 \), resulting in \( z = \pm 5 \) and points \( (0, 0, -8) \) and \( (0, 0, 2) \). Locating these intercepts helps in plotting the accurate position of the sphere within the coordinate system. Understanding this concept is instrumental in fields such as physics and engineering, where intercepts can dictate points of interactions and constraints.

3D Coordinate System

The 3D coordinate system is a framework for pinpointing the location of points in three-dimensional space. It consists of three mutually perpendicular axes: x, y, and z.

Every point in this space can be described by a set of three coordinates \( (x, y, z) \) representing its position along the respective axes. For instance, in the context of the sphere equation \( x^2 + (y+3)^2 + z^2 = 25 \), the center of the sphere is at the point \( (0, -3, 0) \) where the x-coordinate is \( 0 \) since it lies on the yz-plane.

Such a coordinate system is fundamental in geometry and physics as it allows for complex shapes and their movements to be analyzed mathematically. By fully grasping this concept, one can more easily visualize and interpret three-dimensional objects and their interactions.