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Chapter 16: Problem 13

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 2 r \leq z \leq 4\\}$$

Short Answer

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#tag_title# Short Answer The given set in cylindrical coordinates forms a truncated cone with its smaller end at the origin and the larger end at z = 4. The shape is created by constraining the z-coordinate in the inequality $2r \leq z \leq 4$, which changes height as we move along the radial direction, r. The 胃-coordinate ranges from 0 to 2蟺, creating a full revolution around the z-axis.

Step by step solution

01

Understand the given set in cylindrical coordinates

The given set in cylindrical coordinates is: $$ \{(r, \theta, z): 2 r \leq z \leq 4\} $$ The inequalities represent constraints on the z-coordinate depending on the r-coordinate. This means the shape will have a height that changes as we move along the radial direction, r.
02

Analyze the inequality

In cylindrical coordinates, z represents height along the vertical axis. Given the inequality, $$ 2 r \leq z \leq 4, $$ we can identify the relationship between the r-coordinate and z-coordinate. When r = 0, the lower bound of the inequality is 0 (z=0) and, as r increases, z also increases linearly.
03

Sketch the shape based on the inequality

Based on the analysis, we can identify the shape as a truncated cone. We start at the origin (z=0) with r=0. As r increases (move away from the origin), the z-coordinate increases while remaining in the inequality bounds. The 胃-coordinate ranges from 0 to 2蟺, which creates an entire revolution around the z-axis. Hence, the final shape is a truncated cone with a smaller end at the origin and the larger end at z = 4. To draw this, start by drawing the z-axis vertically, and the xy-plane represented by a horizontal plane intersecting the z-axis at z = 0. Sketch a cone shape that starts at the origin and extends upwards until it meets a horizontal plane at z = 4, creating a flat surface at the top.

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