If 8 ft. boards of drywall cost $20 and 12 ft. boards cost $29, what is the minimum price for covering a 16 x 24 room with 12 ft. ceilings (entire room)? 

1. Calculate the surface area to be covered:

Walls:

Perimeter of the room: 2×(16 ft+24 ft)=2×40 ft=80 ft

Area of the walls: 80 ft×12 ft (ceiling height)=960 sq ft

Ceiling:

Area of the ceiling: 16 ft×24 ft=384 sq ft

Total surface area: 960 sq ft + 384 sq ft = 1344 sq ft

2. Determine the area covered by each board type:

Assuming standard drywall width is 4 ft:

8 ft. board: 8 ft×4 ft=32 sq ft

12 ft. board: 12 ft×4 ft=48 sq ft

3. Calculate the cost per square foot for each board type:

8 ft. board: $20/32 sq ft=$0.625 per sq ft

12 ft. board: $29/48 sq ft≈$0.604 per sq ft

The 12 ft. boards are slightly more cost-effective per square foot. To minimize costs, we should prioritize using 12-foot boards as much as possible.

4. Calculate the number of 12 ft. boards needed:

Number of 12 ft. boards = ⌈1344 sq ft/48 sq ft/board⌉=⌈28⌉=28 boards (We use the ceiling function because you can't buy a fraction of a board).

5. Calculate the total cost:

Total cost = 28 boards×$29/board=$812

Therefore, the minimum price for covering the room with drywall is $812.