A rectangular room holds a rectangular carpet in the center of measuring 8m by 16m.. The width form the edge of the carpet to each wall is the same. The area not covered by the carpet is 112m. What is the width of the uncovered area around the carpet?
(8 + 2w) (16 + 2w) = (8 * 16) + 112
dividing by 4 ... (4 + w)(8 + w) = (4*8) + 28
w^2 + 12w + 32 = 60 ... w^2 + 12w - 28 = 0
To find the width of the uncovered area around the carpet, we need to calculate the dimensions of the carpet first and then subtract it from the total dimensions of the room.
Given:
Length of the room = 16m
Width of the room = 8m
Area not covered by the carpet = 112m²
Let's assume that the width of the uncovered area around the carpet is "x" meters.
Now, we can calculate the dimensions of the carpet by subtracting twice the width "x" from both the length and width of the room:
Length of the carpet = Length of the room - 2x
Width of the carpet = Width of the room - 2x
The area of the carpet is the product of its length and width:
Area of the carpet = (Length of the carpet) * (Width of the carpet)
The area not covered by the carpet is equal to the total area of the room minus the area of the carpet:
Area not covered by the carpet = (Length of the room * Width of the room) - (Area of the carpet)
Now, we can substitute the given values and calculate the width of the uncovered area:
112m² = (16m * 8m) - [(16m - 2x) * (8m - 2x)]
Expanding the equation:
112m² = 128m² - (16m * 2x) - (8m * 2x) + (4x²)
Combine like terms:
112m² = 128m² - 32mx - 16mx + 4x²
Rearrange the equation to set it equal to zero:
0 = 128m² - 32mx - 16mx + 4x² - 112m²
Combine like terms:
0 = 16m² - 48mx + 4x²
Now, we have a quadratic equation. To solve it, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = [-b ± √(b² - 4ac)] / (2a)
In our equation, a = 4, b = -48m, and c = 16m².
Substitute the values into the quadratic formula:
x = [-(-48m) ± √((-48m)² - 4 * 4 * 16m²)] / (2 * 4)
Simplify:
x = [48m ± √(2304m² - 256m²)] / 8
x = [48m ± √(2048m²)] / 8
x = (48m ± √(2048) * m) / 8
Simplify the square root:
x = (48m ± 45.254m) / 8
Now, we have two possible solutions:
x = (48m + 45.254m) / 8
x = 93.254m / 8
x ≈ 11.657m
or
x = (48m - 45.254m) / 8
x = 2.746m / 8
x ≈ 0.343m
Therefore, the width of the uncovered area around the carpet can be either approximately 11.657 meters or approximately 0.343 meters, depending on the specific dimensions of the room.