Question $1[30$ pts$]$

The following equation describes the temperature distribution in a flat rectangular metal plate. The temperature on

three sides is held constant at T$1,$ and at T$2$ on the fourth side $($see Figure $1).$ The temperature T $($x$,$ y$)$ as a function

of the xy coordinates shown is given by

T$($x$,$ y$)=($T$2$ T$1)$w$($x$,$ y$)+$ T$1$

where

w$($x$,$ y$)=2$

$\backslash$pi

X$\backslash$infty

n$=$odd

$2$

n

sin

n$\backslash$pi x

L

sinh$($n$\backslash$pi y$/$L$)$

sinh$($n$\backslash$pi W$/$L$)$

The given data for this problem are T$1=70$F$,$ T$2=200$F$,$ and W $=$ L $=2$ ft$.$

Using a spacing of $0\mathrm{.}2$ for both x and y$,$ generate a surface mesh plot and a contour plot of the temperature distribution.

$($Hint: Use two nested for loops, one over x and one over y$.$ Within the y loop, use a while loop $($or use any other

method that you know$)$ to determine how many terms are required in the series to produce a temperature calculation

that is accurate to within $1\%.)$