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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2F9df93f36-08b8-4013-a740-dc24162bd0af%2Ff4xw5a_processed.jpeg&w=3840&q=75)
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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