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## Homework Statement

A slab is in a steady state with temperature T

_{0}at x = 0, and T

_{1}at x = 1. The thermal conductivity is given by K(x) = K

_{0}e

^{[tex]\epsilon[/tex]x}where |[tex]\epsilon[/tex]| << 1. The governing

equation is given by, [tex]\frac{d}{dx}[/tex](K

_{0}e

^{[tex]\epsilon[/tex]x}[tex]\frac{dT}{dx}[/tex]) = 0

(1). Obtain an approximation solution to the temperature distribution by replacing K(x)

with its average value [tex]\bar{K}[/tex] = [tex]\frac{\int K(x) dx}{\int dx}[/tex] over the slab (integrals from 0 to 1)

(2). Otain an exact solution to the temperature distribution.

(3). Rewrite K(x) = K(x) − [tex]\bar{K}[/tex] + [tex]\bar{K}[/tex] then term K(x) − [tex]\bar{K}[/tex] is neglected while replacing K(x) with [tex]\bar{K}[/tex]. Check consistency, i.e, prove that |[tex]\frac{K(x) - \bar{K}}{\bar{K}}[/tex]| << 1

## Homework Equations

## The Attempt at a Solution

Any hints on how to even start?