Determined to reduce the \(\$ 7\) per week cost associated with heat loss through their patio window by convection and radiation, the tenants of Problem 9.15 cover the inside of the window with a 50-mm-thick sheet of extruded insulation. Because they are not very handy around the house, the insulation is installed poorly, resulting in an \(S=5-\mathrm{mm}\) gap between the extruded insulation and the window pane, allowing the room air to infiltrate into the space between the pane and the insulation.

(a) Determine the rate of heat loss through the window and associated weekly cost with the ill-fitting insulation in place. The insulation will significantly reduce the radiation losses through the window. Losses will be due almost entirely to convection.

(b) Plot the rate of heat loss through the patio window as a function of the gap spacing for \(1 \mathrm{~mm} \leq\) \(S \leq 20 \mathrm{~mm}\).