2007 AMC 10B Problem 1
Isabella's house has bedrooms. Each bedroom is feet long, feet wide, and feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy square feet in each bedroom. How many square feet of walls must be painted?
Show solution without answering
Key insight
The four walls unfold into one strip 44 feet long and 8 feet tall. Take away the doors and windows, then multiply by three rooms.
Solution
Only the walls get paint. The floor and ceiling do not, so leave them out.
1. Picture walking once around one bedroom. You pass a -foot wall, a -foot wall, another -foot wall and another -foot wall. That is feet of wall.
2. If you could unfold the four walls and lay them side by side, they would make one long rectangle, feet long and feet tall. Its area is
3. The doors and windows in that room are not painted, so take away their square feet:
4. There are bedrooms that are all the same, so multiply:
The answer is .
Why this works
The walls of a box-shaped room, laid out flat, make one rectangle whose length is the distance around the floor. That saves adding four separate wall areas. Before you compute, read closely which surfaces count, and handle one room completely before multiplying by the number of rooms.
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
Only the four walls of each room are painted, not the floor or ceiling. Unrolled, the four walls form one long rectangle whose width is the perimeter of the floor and whose height is feet:
Remove the doors and windows: square feet per bedroom.
Three bedrooms: .
The answer is .
Review in the AoPS books
- Introduction to Geometry, Chapter 14: Three-Dimensional Geometry
- Introduction to Geometry, Chapter 15: Curved Surfaces
Similar problems
Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site. Also appeared as 2007 12B #1.