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Parametrize the intersection of the surfaces using t = z as the parameter. (If there are multiple correct answers, only enter one.) x2 + y? = z?, 6y = z?

Parametrize the intersection of the surfaces using t = z as the parameter. (If there are multiple correct answers, only enter one.) x2 + y? = z?, 6y = z?

Calculus: Early Transcendentals
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
**Problem Statement:** Parametrize the intersection of the surfaces using \( t = z \) as the parameter. (If there are multiple correct answers, only enter one.) \[ x^2 + y^2 = z^2, \quad 6y = z^2 \] **Solution:** To find the parametrization, express \( x \), \( y \), and \( z \) in terms of \( t \). Given that \( t = z \), we substitute \( t \) for \( z \) in the equations: 1. \( x^2 + y^2 = t^2 \) 2. \( 6y = t^2 \) From the second equation, solve for \( y \): \[ y = \frac{t^2}{6} \] Substitute \( y = \frac{t^2}{6} \) into the first equation: \[ x^2 + \left(\frac{t^2}{6}\right)^2 = t^2 \] \[ x^2 + \frac{t^4}{36} = t^2 \] \[ x^2 = t^2 - \frac{t^4}{36} \] \[ x^2 = \frac{36t^2 - t^4}{36} \] \[ x = \pm \frac{\sqrt{36t^2 - t^4}}{6} \] So the parametrized form is: \[ r(t) = \left( \pm \frac{\sqrt{36t^2 - t^4}}{6}, \frac{t^2}{6}, t \right) \]
Transcribed Image Text:**Problem Statement:** Parametrize the intersection of the surfaces using \( t = z \) as the parameter. (If there are multiple correct answers, only enter one.) \[ x^2 + y^2 = z^2, \quad 6y = z^2 \] **Solution:** To find the parametrization, express \( x \), \( y \), and \( z \) in terms of \( t \). Given that \( t = z \), we substitute \( t \) for \( z \) in the equations: 1. \( x^2 + y^2 = t^2 \) 2. \( 6y = t^2 \) From the second equation, solve for \( y \): \[ y = \frac{t^2}{6} \] Substitute \( y = \frac{t^2}{6} \) into the first equation: \[ x^2 + \left(\frac{t^2}{6}\right)^2 = t^2 \] \[ x^2 + \frac{t^4}{36} = t^2 \] \[ x^2 = t^2 - \frac{t^4}{36} \] \[ x^2 = \frac{36t^2 - t^4}{36} \] \[ x = \pm \frac{\sqrt{36t^2 - t^4}}{6} \] So the parametrized form is: \[ r(t) = \left( \pm \frac{\sqrt{36t^2 - t^4}}{6}, \frac{t^2}{6}, t \right) \]
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