Derive an expression for the temperature distribution

Question

The diagram shows a truncated conical section fabricated from a material of thermal conductivity K. The circular cross-section of the conical section has the diameter D = ax, where a is a constant and x is the axial distance of the section from the apex of the cone. The temperatures at the two end faces of the conical section (at distances x₁ from the apex) are respectively T₁ and T₂ while the lateral surface of the truncated cone is thermally insulted.

Derive an expression for the temperature distribution T(X) in symbolic form assuming one-dimentsional steady-state conditions. Sketch the temperature distribution.

1. Calculate the heat rate q_x through the cone in x-direction.                                            20+40