A furnace is required to sinter powder-metal parts. It operates continuously at \(650^{\circ} \mathrm{C}\) while the parts are fed through on a moving belt. You are asked to select a material for furnace insulation to minimize heat loss and thus to make the furnace as energy-efficient as possible. For reasons of space the insulation is limited to a maximum thickness of \(x=0.2 \mathrm{~m}\).

a. List the function, constraints, objective and free variable.

b. Derive a material index for selecting the material for the insulation.