decorator has 92 one foot square tiles to be laid around edge of 12 x 15 ft room. A rectangular rug that is 3 feet
longer than it is wide is to be placed in center area where there are not tiles.
Find the dimensions of the smallest rug that will cover the untiled part of the floor. All tiles are used and none of them are split
180 - 92 = 88
The rug must be 8 ft by 11 ft.
To find the dimensions of the smallest rug that will cover the untiled part of the floor, we first need to determine the area of the room and the area of the tiled part.
The area of the room can be calculated by multiplying its length by its width:
Area of the room = Length × Width
Area of the room = 12 ft × 15 ft
Area of the room = 180 sq ft
Next, we need to calculate the area covered by the 92 one-foot square tiles. Since each tile is 1 ft × 1 ft, the total area covered by the tiles is:
Area covered by tiles = Number of tiles × Area of each tile
Area covered by tiles = 92 tiles × 1 sq ft
Area covered by tiles = 92 sq ft
To find the area of the untiled part of the floor, we subtract the area covered by the tiles from the total area of the room:
Area of untiled part = Area of the room - Area covered by tiles
Area of untiled part = 180 sq ft - 92 sq ft
Area of untiled part = 88 sq ft
Now, let's consider the rectangular rug that will be placed in the untiled part of the floor. We are told that the rug is 3 feet longer than it is wide. Let's assume the width of the rug is x ft, so its length would be (x + 3) ft.
The area of the rug can be calculated by multiplying its length by its width:
Area of the rug = Length × Width
Area of the rug = (x + 3) ft × x ft
Area of the rug = x^2 + 3x ft^2
We want to find the smallest possible rug that covers the untiled part of the floor, which means we want to find the dimensions that minimize the area of the rug. To do this, we can take the derivative of the area of the rug equation with respect to x and set it equal to zero.
d/dx (Area of the rug) = d/dx (x^2 + 3x)
0 = 2x + 3
Now, solve for x:
2x + 3 = 0
2x = -3
x = -3/2
Since we can't have a negative width for the rug, we discard this solution. Therefore, there is no real number solution for x, which means there is no rectangular rug that can cover the untiled part of the floor with the given conditions.
In other words, there is no smallest rug that can cover the untiled part of the floor using the available tiles.