Find an equation for the surface obtained by rotating the line z = 2x about the z-axis. + x²) b. z? = 4(y2 + x?) c. x² = 4(z² + y²) d. x2 = 4(y² + z²) e. y=2/(x² + z?) a. z=2/(y? %3D

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Question:**

Find an equation for the surface obtained by rotating the line \( z = 2x \) about the z-axis.

**Options:**

a. \( z = 2 \sqrt{y^2 + x^2} \)  
b. \( z^2 = 4(y^2 + x^2) \)  
c. \( x^2 = 4(z^2 + y^2) \)  
d. \( x^2 = 4(y^2 + z^2) \)  
e. \( y = 2 \sqrt{x^2 + z^2} \)

**Answer Choices:**

- ☐ e
- ☐ d
- ☐ b
- ☐ c
- ☐ a

**Explanation of Solutions:**

To solve this problem, recognize that rotating the line \( z = 2x \) about the z-axis will form a surface of revolution, specifically a cone. Transforming the equation to account for the circular symmetry around the z-axis involves:

1. Setting the original line equation, \( z = 2x \), in cylindrical coordinates by replacing \( x \) with \( \sqrt{x^2 + y^2} \).
2. This transformation results in an equation relating \( z \) to the radial distance from the z-axis, which, when squared, can often be simplified to show the geometric shape (such as a cone).
Transcribed Image Text:**Question:** Find an equation for the surface obtained by rotating the line \( z = 2x \) about the z-axis. **Options:** a. \( z = 2 \sqrt{y^2 + x^2} \) b. \( z^2 = 4(y^2 + x^2) \) c. \( x^2 = 4(z^2 + y^2) \) d. \( x^2 = 4(y^2 + z^2) \) e. \( y = 2 \sqrt{x^2 + z^2} \) **Answer Choices:** - ☐ e - ☐ d - ☐ b - ☐ c - ☐ a **Explanation of Solutions:** To solve this problem, recognize that rotating the line \( z = 2x \) about the z-axis will form a surface of revolution, specifically a cone. Transforming the equation to account for the circular symmetry around the z-axis involves: 1. Setting the original line equation, \( z = 2x \), in cylindrical coordinates by replacing \( x \) with \( \sqrt{x^2 + y^2} \). 2. This transformation results in an equation relating \( z \) to the radial distance from the z-axis, which, when squared, can often be simplified to show the geometric shape (such as a cone).
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