Problem 51 - College Algebra and Calculus: An Applied Approach

Question

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 given sphere is a circle on the yz-plane with center at (0,-3) and radius 5.

Step 1: Identifying the Sphere's Center and Radius

The equation of a sphere in standard form is \( (x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2} \), where the center of the sphere is at point (a,b,c) and r is the radius. In our given equation \( x^{2}+(y+3)^{2}+z^{2}=25 \), comparing it with the standard form, one can see that the center of the sphere is at (0,-3,0) and the radius is 5 (since the square root of 25 is 5).

Step 2: Finding the yz-Trace

To find the yz-trace, set x to zero in the equation of the sphere and solve for z. That gives \( (0)^{2}+(y+3)^{2}+z^{2}=25 \) which simplifies the equation to \( (y+3)^{2}+z^{2}=25 \). This is the equation of a circle in the yz-plane.

Step 3: Drawing the yz-Trace

Plot the circle on the yz-plane. The center of this circle is (0,-3), which came from ignoring the x-axis. The radius of the circle is 5, the same as our sphere. Thus the yz-trace of the sphere is a circle centered at (0,-3) on the yz-plane with radius 5.

Key Concepts

Understanding the Center of a Sphere

In geometry, the center of a sphere is a crucial point that is equidistant from all points on the sphere's surface. When you're given the equation of a sphere in the form \((x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}\), the center can be found at the coordinates (a, b, c).

For example, let's look at the equation \(x^{2}+(y+3)^{2}+z^{2}=25\). This can be rewritten as \((x-0)^{2} + (y+3)^{2} + (z-0)^{2} = 5^{2}\). So, we identify the center as (0, -3, 0):

• The x-coordinate is 0.
• The y-coordinate is -3.
• The z-coordinate is 0.

This center tells us where the sphere is positioned in the 3D space.

Determining the Radius of a Sphere

The radius of a sphere is the distance from the center of the sphere to any point on its surface. In the standard equation format \((x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}\), \(r\) is the radius.

Returning to our example, \(x^{2}+(y+3)^{2}+z^{2}=25\) compares to the standard form as \(r^{2}=25\), which gives us \(r = \sqrt{25} = 5\). This indicates the radius is:

• 5 units long.

The sphere extends in all directions from its center, forming a perfect 3D round shape.

Exploring the YZ-Trace of a Sphere

The yz-trace is the intersection of a sphere with the yz-plane. This means we're looking at where the shape of the sphere interacts with this particular two-dimensional plane.

To find it, we set the x variable in the sphere's equation to zero because we're ignoring the x dimension:\[(0)^{2}+(y+3)^{2}+z^{2}=25\]This simplifies to:\((y+3)^{2}+z^{2}=25\).

This new equation represents a circle sitting flat within the yz-plane. It shows that the yz-trace is:

• A circle centered at (0, -3).
• With a radius of 5.

This visualization helps in understanding how 3D objects manifest within 2D spaces.

Circle in Coordinate Plane

In mathematics, circles can often manifest when we slice through 3D objects at a particular plane. Here, in the yz-plane, we explored how a sphere traces out a circle.

The equation \((y+3)^{2}+z^{2}=25\) represents a circle:

• Centered at the point (0, -3) in the yz-plane.
• Having a radius of 5 units.

When visualizing this, imagine looking at a sphere from the side, cutting through it with a flat surface. This resulting circle is a cross-section of the sphere, revealing the relationships between different dimensions.