Chapter 10: Problem 17
Sketch the cylinder in space. \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\)
Short Answer
Expert verified
The cylinder is formed by extruding the ellipse along the z-axis.
Step by step solution
01
Identify the shape's equation
The given equation is \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). This is the equation of an ellipse in the xy-plane with a semi-major axis of 3 along the y-axis and a semi-minor axis of 2 along the x-axis.
02
Understand the concept of a cylinder in 3D space
A cylinder can be thought of as the extrusion of a two-dimensional shape along a third axis. For this problem, extending the two-dimensional ellipse along the z-axis will create a cylindrical shape.
03
Sketch the elliptical base in the xy-plane
Draw an ellipse centered at the origin in the xy-plane, with the major axis of length 6 (2\(\times 3\)) along the y-direction and the minor axis of length 4 (2\(\times 2\)) along the x-direction.
04
Extend the ellipse in the z-direction
Imagine the ellipse being infinitely extended or extruded along the z-axis. This means it maintains its elliptical cross-section while stretching parallel to the z-axis, forming an elliptical cylinder.
05
Sketch the cylinder
Draw parallel elliptical shapes above and below the one in the xy-plane. Use vertical lines to connect corresponding points of the ellipses above and below to represent the extrusion, forming the image of a cylinder in space.
06
Add final touches to the sketch
Make sure to label your axes (x, y, and z) to clearly indicate the orientation. Add a few dashed or faint lines for the backside ellipse to give a sense of depth and complete the sketch of the cylinder.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cylinder
In the realm of 3D geometry, a cylinder is a fascinating shape. Think of it as a long tube stretched endlessly in one direction. Imagine if you took a circle and extended it upwards and downwards, the shape you would get is a circular cylinder.
In our specific case, we are dealing with an elliptical cylinder due to the ellipse equation given in the exercise. An elliptical cylinder takes a similar stretching approach, but instead of stretching a typical circle, you stretch an ellipse. This produces a series of ellipses stacked on top of each other along a third dimension.
The concept of a cylinder reminds us that many 3D shapes can be understood as extensions of 2D shapes. This visualization makes understanding complex geometry much simpler.
In our specific case, we are dealing with an elliptical cylinder due to the ellipse equation given in the exercise. An elliptical cylinder takes a similar stretching approach, but instead of stretching a typical circle, you stretch an ellipse. This produces a series of ellipses stacked on top of each other along a third dimension.
The concept of a cylinder reminds us that many 3D shapes can be understood as extensions of 2D shapes. This visualization makes understanding complex geometry much simpler.
Ellipse
An ellipse is like a squashed or stretched circle. It has two axes which define its shape: the major axis, which is the longest distance across the ellipse, and the minor axis, which is the shortest.
In the given equation, \[\frac{x^2}{4} + \frac{y^2}{9} = 1\],the values underneath the squared variables tell us the lengths of these axes. The major axis, valued at 9 under \(y^2\), indicates it stretches further in the y-direction, specifically 3 units either way—thus 6 units long in total. Likewise, the minor axis, being shorter, stretches b2 units in the x-direction.
Ellipses are important in geometry because they can be transformed into cylinders when extended in the third dimension, like what we are doing in this problem.
In the given equation, \[\frac{x^2}{4} + \frac{y^2}{9} = 1\],the values underneath the squared variables tell us the lengths of these axes. The major axis, valued at 9 under \(y^2\), indicates it stretches further in the y-direction, specifically 3 units either way—thus 6 units long in total. Likewise, the minor axis, being shorter, stretches b2 units in the x-direction.
Ellipses are important in geometry because they can be transformed into cylinders when extended in the third dimension, like what we are doing in this problem.
Extrusion
Extrusion in 3D geometry refers to the technique of extending a 2D figure into the third dimension. Imagine pushing a shape outwards to give it some depth.
In our problem, we take the 2D ellipse and extrude it along the z-axis. This means we stretch the ellipse from the xy-plane infinitely upwards and downwards, creating a shape with parallel elliptical faces all aligned along one direction.
The idea of extrusion allows us to interact with familiar 2D figures and visualize how they behave in a 3D space. This gives us a simple way to understand more complex volumes and structures effectively.
In our problem, we take the 2D ellipse and extrude it along the z-axis. This means we stretch the ellipse from the xy-plane infinitely upwards and downwards, creating a shape with parallel elliptical faces all aligned along one direction.
The idea of extrusion allows us to interact with familiar 2D figures and visualize how they behave in a 3D space. This gives us a simple way to understand more complex volumes and structures effectively.
Z-axis
The z-axis is a crucial part of 3D space. Along with the x-axis and y-axis that we are already familiar with in 2D plane geometry, the z-axis introduces depth.
In our exercise, the z-axis is the axis along which we extend, or extrude, the ellipse to form a cylinder. Just as the y-axis controls up-down movement and the x-axis controls side-to-side movement, the z-axis handles forward-backward movement in 3D modeling.
Being comfortable with manipulating figures along the z-axis helps us more seamlessly transition into working with three-dimensional objects, like cylinders.
In our exercise, the z-axis is the axis along which we extend, or extrude, the ellipse to form a cylinder. Just as the y-axis controls up-down movement and the x-axis controls side-to-side movement, the z-axis handles forward-backward movement in 3D modeling.
Being comfortable with manipulating figures along the z-axis helps us more seamlessly transition into working with three-dimensional objects, like cylinders.
XY-plane
The xy-plane is a fundamental reference in geometry, representing a flat surface where two axes intersect. It helps us determine position on a graph with two dimensions—typically left-right for the x-axis and up-down for the y-axis.
For our problem involving the elliptical cylinder, this plane is where the base ellipse lies before being stretched into the third dimension. The equation given operates within this plane, guiding us to draw the initial 2D shape.
Understanding the xy-plane is essential because it forms the basis for our initial sketches and helps situate our 3D projections. Without a stable understanding of this plane, visualizing 3D structures becomes challenging.
For our problem involving the elliptical cylinder, this plane is where the base ellipse lies before being stretched into the third dimension. The equation given operates within this plane, guiding us to draw the initial 2D shape.
Understanding the xy-plane is essential because it forms the basis for our initial sketches and helps situate our 3D projections. Without a stable understanding of this plane, visualizing 3D structures becomes challenging.
