The length of a rectangular floor is 2 feet more than its width. The area of the floor is 168 square feet. Kim wants to use a rug in the middle of the room and leave a 2 foot border of the floor visible on all sides. What should the length (the longer side) of the rug be?
x(x+2) = 168
x = 12
x + 2 = 14
The longer side should leave 2 feet at each end.
14 - 4 = 10
Step 1: Let's assign variables to the length and width of the rectangular floor.
Let's say the width of the rectangular floor is "w" feet.
Then, the length of the rectangular floor would be "w + 2" feet, as given that the length is 2 feet more than the width.
Step 2: Calculate the area of the rectangular floor.
The area of a rectangle is given by the formula: Area = length * width.
In this case, the area of the floor is given as 168 square feet.
So we can write the equation as: (w + 2) * w = 168.
Step 3: Simplify the equation and solve for 'w'.
Expand the equation: w^2 + 2w = 168.
Step 4: Rearrange the equation to bring all terms to one side and set it equal to zero.
So we get: w^2 + 2w - 168 = 0.
Step 5: Factorize the quadratic equation.
By factoring the equation, we can find the values of 'w' that satisfy this equation.
The factors of -168 that sum up to 2 are 14 and -12.
So, we can write the equation as: (w + 14)(w - 12) = 0.
Step 6: Solve for 'w'.
Set each factor equal to zero and solve for 'w':
w + 14 = 0 or w - 12 = 0.
Solving these equations, we get: w = -14 or w = 12.
Step 7: We discard the negative value of 'w' since the width can't be negative.
Therefore, the width of the rectangular floor is 12 feet.
Step 8: Calculate the length of the rectangular floor.
The length is 2 feet more than the width, so the length would be 12 + 2 = 14 feet.
Step 9: Calculate the length of the rug.
Kim wants to leave a 2-foot border on all sides, so we need to subtract 4 feet from both dimensions (length and width) to get the dimensions of the rug.
The length of the rug would be the length of the floor minus twice the border size, which is 14 - 4 = 10 feet.
Therefore, the length of the rug should be 10 feet.
To find the length of the rug, we first need to determine the dimensions of the rectangular floor.
Let's assume the width of the floor is "x" feet.
According to the given information, the length is 2 feet more than the width, so it would be "x + 2" feet.
The area of the floor is given as 168 square feet. Therefore, we can set up the following equation:
Length × Width = Area
(x + 2) × x = 168
Expanding the equation, we have:
x² + 2x = 168
Rearranging the equation into a quadratic form:
x² + 2x - 168 = 0
To solve this quadratic equation, we can either factorize it or use the quadratic formula. In this case, factoring might be a little challenging, so we will use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values from our equation:
x = (-(2) ± √((2)² - 4(1)(-168))) / (2(1))
Simplifying further:
x = (-2 ± √(4 + 672)) / 2
x = (-2 ± √676) / 2
x = (-2 ± 26) / 2
Now, let's solve for both possibilities:
For x = (-2 + 26) / 2 = 24 / 2 = 12
And x = (-2 - 26) / 2 = -28 / 2 = -14
Since the width cannot be negative, we disregard the negative solution.
Therefore, the width of the floor is 12 feet, and the length is 12 + 2 = 14 feet.
To determine the length of the rug correctly, we need to consider that Kim wants to leave a 2-foot border visible on all sides.
So the length of the rug would be the length of the floor minus twice the width of the border.
Length of the rug = 14 - (2 × 2) = 14 - 4 = 10 feet.
Therefore, the length of the rug should be 10 feet.