A 4m by 6m rug covers half of the floor area of a room and leaves a uniform strip of bare floor around the edges. What are the dimensions of the room?
To find the dimensions of the room, we can subtract the area covered by the rug from the total area of the room.
First, we need to calculate the area covered by the rug. The area of a rectangle is calculated by multiplying its length by its width. In this case, the rug is 4 meters by 6 meters, so its area is 4 * 6 = 24 square meters.
Since the rug covers half of the floor area, the total area of the room can be obtained by multiplying the rug's area by 2. So, the total area of the room is 24 * 2 = 48 square meters.
Now, let's assume the room's dimensions as length (L) and width (W). We know that the rug leaves a uniform strip of bare floor around the edges, which means, both the length and width dimensions of the room need to be greater than the corresponding rug dimensions. Since the rug is 4 meters by 6 meters, we can express the room's dimensions as:
Length (L) = 4 + x
Width (W) = 6 + y
Now, we can calculate the remaining area surrounding the rug, which is the difference between the total area of the room and the area covered by the rug. The remaining area can be expressed as:
Remaining area = L * W - Rug area
Remaining area = (4 + x) * (6 + y) - 24
Since the remaining area leaves a uniform strip around the rug, the equation can be set up as:
Remaining area = 2x(x + y)
We know that the remaining area is half of the total area of the room, so we can set it equal to half of the room's area (48 / 2 = 24 square meters). Therefore:
2x(x + y) = 24
Simplifying the equation, we have:
2x^2 + 2xy = 24
Now, we can try different values for x and y until we find a valid solution for the equation. Let's start with a value of x = 2:
2(2)^2 + 2(2)y = 24
8 + 4y = 24
4y = 24 - 8
4y = 16
y = 16 / 4
y = 4
So, when x = 2 and y = 4, the equation is valid. This means that the room's dimensions are 4 + 2 = 6 meters for the length and 6 + 4 = 10 meters for the width.
Therefore, the dimensions of the room are 6 meters (length) by 10 meters (width).