The temperature at a point (x,y,z) of a solid E bounded by the coordinate planes and the plane x + y + z = 1 is T(x, y, z) = (xy + 8z − 8) degrees Celsius. Find the average temperature over the solid. (Answer to 3 decimal places). Average Value of a function using 3 variables N y 0 X
The temperature at a point (x,y,z) of a solid E bounded by the coordinate planes and the plane x + y + z = 1 is T(x, y, z) = (xy + 8z − 8) degrees Celsius. Find the average temperature over the solid. (Answer to 3 decimal places). Average Value of a function using 3 variables N y 0 X
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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5.4.9
Transcribed Image Text:**Transcription and Explanation for Educational Website**
**Text Description:**
The temperature at a point \((x,y,z)\) of a solid \(E\) bounded by the coordinate planes and the plane \(x + y + z = 1\) is \(T(x, y, z) = (xy + 8z - 8)\) degrees Celsius. Find the average temperature over the solid. (Answer to 3 decimal places).
**Graph Description:**
The graph shown is a 3D plot illustrating the solid \(E\) within a cube bounded by the coordinate planes and the plane \(x + y + z = 1\). The axes labeled \(x\), \(y\), and \(z\) range from 0 to 1.
- The plane \(x + y + z = 1\) forms a triangular face intersecting the \(x\), \(y\), and \(z\) axes at the point where each of these coordinates equals 1, creating a right-angled triangular region.
- The solid beneath this triangular plane (highlighted in green) represents the volume \(E\) for which we need to find the average temperature.
- The graph's title is "Average Value of a function using 3 variables," indicating the focus on finding the average value over the specified region in three-dimensional space.
This visualization aids in understanding the boundaries and region of integration needed to solve the average temperature problem for the given mathematical function.
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