Question

56. DIG DEEPER! Two rectangular, adjacent rooms share a

wall. One-foot-by-one-foot tiles cover the floor of each room.
Describe how the greatest possible length of the adjoining
wall is related to the total number of tiles in each room. Draw
a diagram that represents one possibility.
pressions and Factors

Answer 1

The greatest length of an adjoining wall shared by two rectangular rooms covered by one-foot square tiles is maximized when one dimension of each room is the same. This dimension would be equal to the length of the shared wall. To ensure a large shared wall length, one room could be narrow while the other is wide, or both rooms could have matching dimensions.

Step-by-step explanation:

The answer to this question involves understanding the relationship between the dimensions of a room and the number of tiles used to cover its floor when the tiles are a standard size. If two rectangular rooms share a wall, the greatest possible length of the adjoining wall will occur when one dimension of each room is the same and maximized.

Consider two adjacent rectangular rooms, A and B, with a common wall between them. Let's say room A is a tiles long and b tiles wide, while room B is c tiles long and b tiles wide. Here, b represents the shared wall length. Since the tiles are one-foot squares, this is also the length of the common wall in feet. The total number of tiles in room A will be a x b, and for room B, it will be c x b.

To maximize the length of the shared wall (b), we could either have one of the two adjoining rooms as narrow as possible (minimizing a or c) and the other one as wide as possible, or have both rooms with identical dimensions, hence maximizing b consistently for both rooms. In this scenario, the total number of tiles for the two rooms would depend on the respective lengths a and c of the other sides of the rooms.

Example Diagram:

• Room A: 4 tiles by 3 tiles (12 tiles in total)
• Room B: 6 tiles by 3 tiles (18 tiles in total)
• Shared wall: 3 tiles (or 3 feet) in length

Answer 2

Consider two adjacent rectangular rooms having Length=L, and, Breadth = B

Now Suppose the wall which is in between two rooms has a height or length =H.

Breadth of wall = B [ if the wall doesn't exceed the breadth of room]

Considering two rooms to be identical,

Area of each room= L × B square unit

Area of each tile = 1×1=1 square unit

Number of tiles required= L B ÷ 1= LB tiles( product of length and breadth of room is number of tiles required)

Suppose if,LB= N

B= N/L .................(1)

Area of wall(W) = B×H= B H square unit

B =W/H ......................(2)

Equating (1) and (2)

⇒N/L = W/ H

⇒H =w-z/N

⇒H =WL/LH

⇒H = W/B

⇒ H =area of wall / Breadth of room and wall