What is the shortest distance from the surface xy + 9x + z² = 81 to the origin? distance =
What is the shortest distance from the surface xy + 9x + z² = 81 to the origin? distance =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
What is the shortest distance from the surface \( xy + 9x + z^2 = 81 \) to the origin?
**Solution:**
distance = [Insert the correct distance here]
### Explanation
To find the shortest distance from the surface defined by the equation \( xy + 9x + z^2 = 81 \) to the origin, we typically use methods from multivariable calculus, such as Lagrange multipliers or gradient descent. The solution involves identifying a point \((x, y, z)\) on the surface that minimizes the Euclidean distance to the origin. The Euclidean distance \(d\) from a point \((x, y, z)\) to the origin is given by:
\[ d = \sqrt{x^2 + y^2 + z^2} \]
You will need to:
1. Set up the function to minimize, in this case, \(f(x, y, z) = x^2 + y^2 + z^2\).
2. Employ the constraint \(g(x, y, z) = xy + 9x + z^2 - 81 = 0\).
3. Use the method of Lagrange multipliers to find the coordinates \((x, y, z)\) satisfying the constraint that also minimize the distance function.
4. Substitute those coordinates back into the distance function to find the minimum distance \(d\).
This mathematical problem involves advanced calculus techniques best understood with some background in multivariable functions and calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe70859f-d1ee-4d19-93f5-d3c21b44393e%2F48b65bab-e38f-4d63-819d-0128738defe0%2F14h5n1_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
What is the shortest distance from the surface \( xy + 9x + z^2 = 81 \) to the origin?
**Solution:**
distance = [Insert the correct distance here]
### Explanation
To find the shortest distance from the surface defined by the equation \( xy + 9x + z^2 = 81 \) to the origin, we typically use methods from multivariable calculus, such as Lagrange multipliers or gradient descent. The solution involves identifying a point \((x, y, z)\) on the surface that minimizes the Euclidean distance to the origin. The Euclidean distance \(d\) from a point \((x, y, z)\) to the origin is given by:
\[ d = \sqrt{x^2 + y^2 + z^2} \]
You will need to:
1. Set up the function to minimize, in this case, \(f(x, y, z) = x^2 + y^2 + z^2\).
2. Employ the constraint \(g(x, y, z) = xy + 9x + z^2 - 81 = 0\).
3. Use the method of Lagrange multipliers to find the coordinates \((x, y, z)\) satisfying the constraint that also minimize the distance function.
4. Substitute those coordinates back into the distance function to find the minimum distance \(d\).
This mathematical problem involves advanced calculus techniques best understood with some background in multivariable functions and calculus.
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