Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}=169\) (a) \(x=5\) (b) \(y=12\)

Step-by-Step Solution

Answer

The trace of the intersection of plane \(x=5\) with the sphere is a circle of radius 12 and centered at (5,0,0). The trace of the intersection of plane \(y=12\) with the sphere is a circle of radius 5 centered at (0,12,0).

Step 1: Determine Intersection Points for \n\(x=5\)

To determine where the plane \(x=5\) intersects the sphere, substitute the value of x into the equation of the sphere. Therefore equation becomes \(5^2 + y^2 + z^2 = 169\). This simplifies to \(y^2 + z^2 = 144 = 12^2\), which represents a circle of radius 12 centered at (5,0,0). This is the trace of the intersection of the plane \(x=5\) with the sphere.

Step 2: Determine Intersection Points for \n\(y=12\)

Next, to find where the plane \(y=12\) intersects the sphere, substitute the value of y into the equation of the sphere. The equation then becomes \(x^2 + 12^2 + z^2 = 169\). This simplifies to \(x^2 + z^2 = 25 = 5^2\), which is the equation of a circle of radius 5 centered at (0,12,0). This is the trace of the intersection of the plane \(y=12\) with the sphere.

Step 3: Sketch the Traces

The final step is to sketch these traces. The trace for the plane \(x=5\) is a circle of radius 12 centered at (5,0,0) on the x-plane. Similarly, the trace for the plane \(y=12\) is a circle of radius 5 centered at (0,12,0) on the y-plane. The sketch would display these traces.

Key Concepts

Trace of Intersection

When studying the intersection of planes with spheres, think of it as cutting through a 3D ball with a flat sheet. This cut, or intersection, results in a shape on the surface of the sphere, often a circle. The exact size and position of the circle depend on the plane's position.

Let's first tackle the trace for the plane defined by \(x = 5\). Substitute \(x = 5\) into the spherical equation \(x^2 + y^2 + z^2 = 169\). The equation simplifies to \(25 + y^2 + z^2 = 169\), or \(y^2 + z^2 = 144\). This tells us that the trace intersects at a circle of radius 12, centered at the point (5,0,0).

For the plane \(y = 12\), we substitute to get \(x^2 + 144 + z^2 = 169\), simplifying to \(x^2 + z^2 = 25\). Here, the trace is a circle with a radius of 5, centered at (0,12,0). This is the core idea of a trace of intersection—it's observing the shapes that appear when slicing through 3D objects with 2D surfaces.

Equation of Sphere

Understanding the equation of a sphere is key to solving intersection problems. A sphere centered at the origin is represented by the formula \(x^2 + y^2 + z^2 = r^2\). Here, \(r\) represents the sphere's radius.

In our exercise, the sphere's equation is \(x^2 + y^2 + z^2 = 169\). By comparing this to the standard form, we recognize that the radius \(r\) is the square root of 169, which equals 13. Knowing the radius helps identify the size of any resulting intersection traces, such as the circles found in this exercise.

• Equations transform: When intersected by planes, parts of the sphere's equation change to reflect only the dimensions involved in the intersection.
• Radius role: The radius indicates how far from the center every point on the sphere's surface is. This is also pivotal when determining the characteristics of intersection circles.

Coordinate Geometry

Coordinate geometry allows us to describe geometric figures and their positions mathematically. It goes beyond simple shapes to explaining how forms like planes and spheres interact in 3D space.

In the context of this exercise, the interplay between the sphere and the planes demonstrates how coordinates help us find intersection points. For instance:

• The coordinate (5,0,0) is important for the plane \(x = 5\), indicating the circle's center formed by the plane intersecting the sphere.
• Similarly, (0,12,0) is the center for the intersection on plane \(y = 12\).

Coordinate geometry simplifies these complex interactions by using algebraic forms. This systematized approach makes it easier to visualize and calculate real-world geometric problems.