75% R38. 25% uninsulated.
1/(Reff + 3) = .25/3 + .75/41
Reff = 7
Or, is it even that good? The number 3 in math so far, represents the sum of convection and conduction resistances in a complex wall with at least two convection interfaces, aiming for a highest number, so as to not over-estimate savings with correction. What if the real sum here is only 1.3?
0.17 + 0.45 + 0.68 = 1.3 (Colorado Energy numbers)
1/(Reff + 1.3) = .25/1.3 + .75/(38 + 1.3)
Reff = 3.4, doubling the heat cost as-found. Ueff = Reff = 1/3.4 = 0.29
My insulation math at 95% furnace efficiency and natural gas cost of $2 per therm, says heat cost per year is: $2.4*Area*.88/.95*Ueff.
Area = 3.6 sq ft.
Cost per year: $2.30. Compare to cost of $0.20 per year at 3.6 sq ft of R38 ceiling, U = 1/39.3.
With my usual analysis with that number 3, Reff = 7, cost per year is:
$2.4*3.6*.88/.95/(7 + 3) = $0.80.
Save $2.10 per year or $1.50 per year with the better hatch cover. Motivation is the same whatever the math. The gob is even more a nuisance and safety hazard, than a false economy. Avoid spending more for the better hatch.
Here is the replacement hatch cover, ready for application of facing drywall. An R19 unfaced batt was at hand, costs less than R15 and is acceptable where it overfills the space.
Here is the completed useful attic with the new hatch cover stowed upon a tray.
I intended to use the 1x4 wood frame in the ceiling as-found. Finding that frame unacceptably out-of-square, a repair was instructive The RHS member of the frame was long by 7/16". In the correction by steel-cutting Sawzall, a steel edge frame for drywall was confirmed. The drywall and steel edging formed into the hole reduced possible opening size by more than an inch.
I think this is a much better arrangement of a generally-useful hatch, allowing maximum hole size with 24" framing on-center tolerance of plus or minus 1/2". It will be better that the hatch and its frame are built with precision as a matched set. I think of ways to increase durability while reducing cost.
Flexible grout completes the ceiling drywall edge over the hatch frame, with texture match. Think to offer a white melamine plywood facing instead of heavy drywall. In practice, a manufactured hatch of standard size will cost less than drywaller steel-edging of a plunker hole while complying with raised energy-efficiency code requirements.