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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 + 0) degrees Celsius. Find the average temperature over the solid. (Answer to 3 decimal places). Average Value of a function using 3 variables N 1- 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 + 0) degrees Celsius. Find the average temperature over the solid. (Answer to 3 decimal places). Average Value of a function using 3 variables N 1- 0 X

Advanced Engineering Mathematics
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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**Title: Average Value of a Function Using Three Variables** **Problem Statement:** The temperature at a point \((x, y, z)\) of a solid \(E\) is given. The solid is bounded by the coordinate planes and the plane \(x + y + z = 1\). The temperature function is defined as: \[ T(x, y, z) = (xy + 8z + 0) \] This represents the temperature in degrees Celsius. The task is to find the average temperature over the solid. The final answer should be rounded to three decimal places. **Explanation of Diagram:** The 3D graph illustrates the region of the solid \(E\). The graph is bounded by: - The coordinate planes: These include the planes defined by \(x = 0\), \(y = 0\), and \(z = 0\). - Plane \(x + y + z = 1\): This is the slanted plane shown intersecting the coordinate axes at the point where \(x + y + z = 1\). The solid within these boundaries is represented visually by the shaded green area in the graph. **Visual Details:** - **Axes:** - The \(x\)-axis ranges from 0 to 1. - The \(y\)-axis ranges from 0 to 1. - The \(z\)-axis is also scaled from 0 to 1. - **Shaded Region:** - The shaded area (in green) signifies the volume of the solid \(E\) where the integration will take place to find the average temperature. This setup provides a clear visual interpretation of the problem, illustrating the boundaries within which the temperature varies, allowing for the computation of an average value.
Transcribed Image Text:**Title: Average Value of a Function Using Three Variables** **Problem Statement:** The temperature at a point \((x, y, z)\) of a solid \(E\) is given. The solid is bounded by the coordinate planes and the plane \(x + y + z = 1\). The temperature function is defined as: \[ T(x, y, z) = (xy + 8z + 0) \] This represents the temperature in degrees Celsius. The task is to find the average temperature over the solid. The final answer should be rounded to three decimal places. **Explanation of Diagram:** The 3D graph illustrates the region of the solid \(E\). The graph is bounded by: - The coordinate planes: These include the planes defined by \(x = 0\), \(y = 0\), and \(z = 0\). - Plane \(x + y + z = 1\): This is the slanted plane shown intersecting the coordinate axes at the point where \(x + y + z = 1\). The solid within these boundaries is represented visually by the shaded green area in the graph. **Visual Details:** - **Axes:** - The \(x\)-axis ranges from 0 to 1. - The \(y\)-axis ranges from 0 to 1. - The \(z\)-axis is also scaled from 0 to 1. - **Shaded Region:** - The shaded area (in green) signifies the volume of the solid \(E\) where the integration will take place to find the average temperature. This setup provides a clear visual interpretation of the problem, illustrating the boundaries within which the temperature varies, allowing for the computation of an average value.
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