Problem 7
Question
In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ r^{2}+z^{2}=4 $$
Step-by-Step Solution
Verified Answer
The graph is a sphere centered at the origin with radius 2.
1Step 1: Identify the Surface Type
The given equation in cylindrical coordinates is \( r^2 + z^2 = 4 \). In cylindrical coordinates, \( r \) represents the radial distance from the z-axis. The equation resembles the equation of a cylinder or sphere in Cartesian coordinates. In this case, it is actually the equation of a cylinder since there are two variables squared and summed.
2Step 2: Convert to Cartesian Coordinates
In cylindrical coordinates, \( r^2 = x^2 + y^2 \). Therefore, substituting \( r^2 \) in the initial equation, we convert it to Cartesian coordinates: \( x^2 + y^2 + z^2 = 4 \). This is the equation of a sphere in three-dimensional space.
3Step 3: Interpret the Equation
The equation \( x^2 + y^2 + z^2 = 4 \) represents a sphere centered at the origin (0,0,0) with a radius of 2, because the equation can be rewritten as \( (x-0)^2 + (y-0)^2 + (z-0)^2 = 2^2 \).
4Step 4: Sketch the Surface
To sketch the graph of the equation in three-dimensional space, draw a sphere centered at the origin with radius 2. The sphere should extend equally in all directions: positive and negative x, y, and z. Use reference points like (2,0,0), (0,2,0), (0,0,2), (-2,0,0), (0,-2,0), and (0,0,-2) to guide your sketch.
Key Concepts
Three-Dimensional SpaceSphereCartesian CoordinatesGraph Sketching
Three-Dimensional Space
Three-dimensional space can be thought of as the world we live in. It is a vast realm where every point is defined using three coordinates typically denoted by \(x, y, \, and \, z\), known as the x-axis, y-axis, and z-axis respectively. These axes intersect each other at the origin (0,0,0). The z-axis adds depth, allowing us to explore heights and depths beyond the simple width and length of two-dimensional space.
This space is essential for visualizing objects that have volume, like spheres, cylinders, and boxes. In this exercise, understanding three-dimensional space helps us visualize the sphere represented by the equation \(x^{2} + y^{2} + z^{2} = 4\). This sphere occupies space in all three directions, making it a fully three-dimensional object.
This space is essential for visualizing objects that have volume, like spheres, cylinders, and boxes. In this exercise, understanding three-dimensional space helps us visualize the sphere represented by the equation \(x^{2} + y^{2} + z^{2} = 4\). This sphere occupies space in all three directions, making it a fully three-dimensional object.
Sphere
A sphere in geometry is a perfectly symmetrical object where every point on its surface is equidistant from its center. You can imagine a sphere like a basketball, perfectly round and without any edges or vertices. Spheres are a perfect example of objects in three-dimensional space.
In this exercise, the equation \((x - 0)^{2} + (y - 0)^{2} + (z - 0)^{2} = 2^{2}\) describes a sphere centered at the origin (0,0,0) with a radius of 2 units. The equation ensures that whatever point you choose on the sphere, it maintains a consistent distance of 2 from the center.
To visualize this, consider slicing an orange; each circular slice represents a section of the sphere. The entirety of these slices forms the whole sphere.
In this exercise, the equation \((x - 0)^{2} + (y - 0)^{2} + (z - 0)^{2} = 2^{2}\) describes a sphere centered at the origin (0,0,0) with a radius of 2 units. The equation ensures that whatever point you choose on the sphere, it maintains a consistent distance of 2 from the center.
To visualize this, consider slicing an orange; each circular slice represents a section of the sphere. The entirety of these slices forms the whole sphere.
Cartesian Coordinates
Cartesian coordinates are a standardized way of describing locations in space using three dimensions: \(x, y, \, and \, z\). They are particularly useful for finding precise locations and understanding the shape and size of objects within space.
Originally, the equation \(r^{2} + z^{2} = 4\) was given in cylindrical coordinates. To understand it deeply, we converted it to Cartesian coordinates: \(x^{2} + y^{2} + z^{2} = 4\). This transformation helps in describing the position of various points in a three-dimensional space and makes it easier to identify that the equation forms a sphere.
Originally, the equation \(r^{2} + z^{2} = 4\) was given in cylindrical coordinates. To understand it deeply, we converted it to Cartesian coordinates: \(x^{2} + y^{2} + z^{2} = 4\). This transformation helps in describing the position of various points in a three-dimensional space and makes it easier to identify that the equation forms a sphere.
- \(x\): The horizontal distance from the y-axis
- \(y\): The horizontal distance from the x-axis
- \(z\): The vertical distance or elevation from the x-y plane
Graph Sketching
Graph sketching involves drawing representations of mathematical equations on paper or digitally to better understand their shapes and spatial relationships. In three-dimensional space, sketching can be more challenging as depth must be communicated on a flat surface.
To sketch the sphere defined by \(x^{2} + y^{2} + z^{2} = 4\), begin by marking the center point at the origin (0,0,0) of your graph. Then, extend the sphere radius equally in every direction. Use points such as (2,0,0), (0,2,0), (0,0,2), (-2,0,0), (0,-2,0), and (0,0,-2) to provide guidance for the boundaries of the sphere.
To aid in visualizing the sphere:
To sketch the sphere defined by \(x^{2} + y^{2} + z^{2} = 4\), begin by marking the center point at the origin (0,0,0) of your graph. Then, extend the sphere radius equally in every direction. Use points such as (2,0,0), (0,2,0), (0,0,2), (-2,0,0), (0,-2,0), and (0,0,-2) to provide guidance for the boundaries of the sphere.
To aid in visualizing the sphere:
- Draw light reference circles horizontally and vertically
- Ensure the sphere looks like it's popping out of the page; shading can help with this effect
Other exercises in this chapter
Problem 7
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