Describe and sketch the surface in 3 dimensional R^3 represented by the equation x^2 +z^2 =9
Notice that there is no y term, so y can be any value
x^2 + z^2 = 9 is a circle in the xz plane with a radius of 3
We are then "pushing this along the y-axis, resulting in a cylinder with radius 3
carefully type in (x^2) + (z^2) = 9 into
https://www.geogebra.org/3d?lang=en
to see the cylinder.
Well, imagine you have a giant donut floating in space. Picture it clearly in your mind: it's circular, delicious, and definitely out of this world. Now, imagine taking a knife and cutting it in half, vertically. What you're left with is a beautiful shape that we call a torus.
Now, this torus shape can be represented mathematically in 3-dimensional space, denoted as R^3. The equation x^2 + z^2 = 9 precisely describes this surface. If you were to plot it on a graph, you would see a hollow tube-like figure with a radius of 3 units, centered at the origin (0,0,0).
But hey, words can only do so much justice to the donut, so here's my slightly questionable sketch of the torus:
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The equation x^2 + z^2 = 9 represents a surface in 3-dimensional space, specifically in the Cartesian coordinate system R^3. This surface is a cylinder oriented along the y-axis, with a radius of 3.
To sketch this surface, start by imagining the xy-plane as the base of your sketch. Then, draw a vertical line in the middle to represent the y-axis. Next, draw a circle centered at the origin (0, 0) on the xy-plane with a radius of 3. This circle represents the base of the cylinder. Finally, extend this circle upwards and downwards indefinitely along the y-axis to complete the sketch of the surface.
The equation x^2 + z^2 = 9 represents a surface in 3-dimensional space, specifically in the Cartesian coordinate system R^3.
To visualize this surface, let's start with the equation x^2 + z^2 = 9. This equation represents a circle with a radius of 3, centered at the origin in the x-z plane (where y = 0). Since the equation does not contain a y term, this means that the y coordinate can take any value.
Now, we can represent this surface by considering the three variables x, y, and z. The variable y does not have any restrictions, so it can take any value. For each value of y, the equation x^2 + z^2 = 9 represents a circle in the x-z plane.
To sketch this in 3-dimensional space, we can start by drawing the x-z plane as a flat surface. On this plane, we can draw the circle with a radius of 3, centered at the origin. This circle lies parallel to the y-axis.
To represent the entire surface, we need to extend this circle along the y-axis. This means that for each possible value of y, there will be a separate circle in the x-z plane with a radius of 3 and centered at the origin. These circles will stack on top of each other infinitely along the y-axis.
In summary, the graph of the equation x^2 + z^2 = 9 in 3-dimensional space is a cylindrical surface, where each cross-section is a circle with a radius of 3. The center of the circles lies on the y-axis, and they extend infinitely in the positive and negative y directions.