Due to a spatial variation in purity, the thermal conductivity of a metal bar (cross sectional area `4xx10^(-4)m^(2)`, length 1m) decreases linearly along its length from 400 `Wm^(-1)K^(-1)` at one end, to 200 `Wm^(-1)K^(-1)` at the other. Calculate the rate at which heat flows through the bar if the hot end is maintained at `200^(@)C` and the cold end at `0^(@)C`

Explanation

To solve the problem step by step, we will follow the outlined reasoning and calculations from the video transcript.

### Step 1: Define the Problem
We have a metal bar with a cross-sectional area \( A = 4 \times 10^{-4} \, m^2 \) and a length \( L = 1 \, m \). The thermal conductivity \( k \) decreases linearly from \( 400 \, Wm^{-1}K^{-1} \) at one end to \( 200 \, Wm^{-1}K^{-1} \) at the other end. The temperatures at the ends are \( T_1 = 200^\circ C \) (hot end) and \( T_2 = 0^\circ C \) (cold end).

### Step 2: Establish the Linear Variation of Thermal Conductivity
The thermal conductivity \( k \) can be expressed as a linear function of \( x \):
\[
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