Question

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}=25\) (a) \(z=3\) (b) \(x=4\)

Short answer

The intersection of the sphere with the plane \(z=3\) will give us a circle of radius 4 centered at the origin, and the intersection of the sphere with the plane \(x=4\) will give us circle of radius 3 centered at the origin.

Step-by-step solution

Find the intersection with the plane \(z=3\)

We can substitute \(z = 3\) into the equation of the sphere:\(x^{2} + y^{2} + 3^{2} = 25\). This will give us the equation of the circle of intersection, namely \(x^{2} + y^{2} = 16\). So, the intersection of the sphere with the plane \(z=3\) is a circle of radius 4 with a center at the origin.

Find the intersection with the plane \(x=4\)

We can substitute \(x = 4\) into the equation of the sphere: \(4^{2} + y^{2} + z^{2} = 25\). This will give us the equation of the circle of intersection, namely \(y^{2} + z^{2} = 9\). So, the intersection of the sphere with the plane \(x=4\) is a circle of radius 3 with a center at the origin.

Sketch the circles

In the plane \(z=3\), draw a circle centered at the origin with a radius of 4 and in the plane \(x=4\), draw a circle centered at the origin with a radius of 3. This sketches represent the intersections of the sphere with the given planes.

Key concepts

Analytic Geometry

Analytic Geometry is a branch of mathematics that uses algebraic equations to describe geometric objects. It combines algebra and geometry to solve problems involving shapes and spaces. Here, we explore how geometric objects like planes and spheres intersect each other using math. Analytic Geometry helps us

• Visualize the relationships between different geometric figures.
• Translate geometric problems into algebraic equations.
• Use these equations to find precise solutions to problems of intersection.

With analytic geometry, one can describe a sphere and plane using equations in a coordinate system and determine where they intersect. This technique is essential for solving complex geometrical problems in physics, engineering, and computer graphics.

3D Coordinate System

A 3D Coordinate System is a space where each point is determined by three coordinates, representing width, depth, and height. The axes in this system are usually labeled as the x-axis, y-axis, and z-axis.

• The sphere in our exercise, with its equation given by \(x^2 + y^2 + z^2 = 25\), exists in this 3-dimensional space.
• Each point on the sphere's surface could be an intersection point with any plane in this system.

In this case, the sphere is centered at the origin o(0, 0, 0) with a radius of 5, because \(25 = 5^2\).The position of other geometric objects, like planes, is determined using similar equations in this space.By understanding the 3D coordinate system, students can easily find and visualize how planes might slice through spheres, creating shapes of lower dimensions, like circles, based on the plane's position in the coordinate space.

Plane and Sphere Intersection

The concept of Plane and Sphere Intersection is about finding the curve which results from the intersection of a plane with a sphere. When a plane cuts through a sphere, a circumferential shape, usually a circle, is often the result.

• In our exercise, substituting values for \(z\) and \(x\) in the equation of the sphere simplifies the problem to a 2D equation of a circle.
• With the plane \(z=3\), we substitute into the sphere equation: \(x^2 + y^2 + 3^2 = 25\), reducing it to \(x^2 + y^2 = 16\). This is a circle with radius 4 centered at the origin.
• For the plane \(x=4\), substituting in the sphere equation: \(4^2 + y^2 + z^2 = 25\), simplifies to \(y^2 + z^2 = 9\), leading to a circle of radius 3 centered at the origin.

Understanding these intersections through algebraic manipulation helps visualize and sketch the result, demonstrating how a flat plane interacts with a curved surface like a sphere.