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Find the shortest Distance d faumthe point to the plane X+y+Z=8.

Find the shortest Distance d faumthe point to the plane X+y+Z=8.

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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The image contains handwritten mathematical text with the following content, formatted appropriately for an educational website: --- **Problem Statement:** Find the shortest distance, \( d \), from the point \( (3, 1, -8) \) to the plane given by the equation \( x + y + z = 8 \). **Solution:** The distance \( d \) from a point \((x_1, y_1, z_1)\) to a plane defined by \( Ax + By + Cz + D = 0 \) can be calculated using the formula: \[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \] In this problem: - The point is \( (x_1, y_1, z_1) = (3, 1, -8) \) - The plane is \( x + y + z - 8 = 0 \), so \( A = 1 \), \( B = 1 \), \( C = 1 \), and \( D = -8 \) Plug the values into the formula to find \( d \). --- *Note: This is a transcription and explanation based on the image and is intended for educational purposes.*
Transcribed Image Text:The image contains handwritten mathematical text with the following content, formatted appropriately for an educational website: --- **Problem Statement:** Find the shortest distance, \( d \), from the point \( (3, 1, -8) \) to the plane given by the equation \( x + y + z = 8 \). **Solution:** The distance \( d \) from a point \((x_1, y_1, z_1)\) to a plane defined by \( Ax + By + Cz + D = 0 \) can be calculated using the formula: \[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \] In this problem: - The point is \( (x_1, y_1, z_1) = (3, 1, -8) \) - The plane is \( x + y + z - 8 = 0 \), so \( A = 1 \), \( B = 1 \), \( C = 1 \), and \( D = -8 \) Plug the values into the formula to find \( d \). --- *Note: This is a transcription and explanation based on the image and is intended for educational purposes.*
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