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Chapter 21: Problem 7

Using the temperature distribution function \(T(x, y)=\frac{\sinh \pi y \sin \pi x}{\sinh \pi}\) in Example 1.6, evaluate the temperature at the centre of the plate.

Short Answer

Expert verified
Answer: The temperature at the center of the square plate is given by the expression \(\frac{\sinh\left(\frac{\pi}{2}\right)}{\sinh(\pi)}\).

Step by step solution

01

Identify the temperature distribution function

The given temperature distribution function is: \(T(x, y) = \frac{\sinh (\pi y) \sin(\pi x)}{\sinh(\pi)}\).
02

Determine the center of the plate

Since the plate is a square with side length 1, the center of the plate will have the coordinates \((0.5, 0.5)\).
03

Substitute the coordinates of the center into the function

Now, we substitute the values of \(x = 0.5\) and \(y = 0.5\) into the temperature distribution function: \(T(0.5, 0.5) = \frac{\sinh (\pi (0.5)) \sin(\pi (0.5))}{\sinh(\pi)}\)
04

Evaluate the expression

\[T(0.5, 0.5) = \frac{\sinh\left(\frac{\pi}{2}\right) \sin\left(\frac{\pi}{2}\right)}{\sinh(\pi)}\] Using the fact that \(\sin\left(\frac{\pi}{2}\right) = 1\): \[T(0.5, 0.5) = \frac{\sinh\left(\frac{\pi}{2}\right)}{\sinh(\pi)}\]
05

Simplify the expression

The temperature at the center of the plate is given by: \[T(0.5, 0.5) = \frac{\sinh\left(\frac{\pi}{2}\right)}{\sinh(\pi)}\] This is the final expression for the temperature at the center of the plate.

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