The length of a rectangle is 1ft more than twice its width, and the area of the rectangle is 28ft^2 . Find the dimensions of the rectangle.
l=
w=
w = width
L = 2w + 1
A = Lw
28 = w(2w + 1)
28 = 2w^2 + w
Solve for w by completing the square.
4 7
To solve this problem, we can use the formula for the area of a rectangle:
Area = length × width
Since we know that the area is 28ft^2, we can substitute this value into the formula and solve for the length.
28 = (2w + 1) × w
Let's simplify this equation:
28 = 2w^2 + w
Rearranging the equation:
2w^2 + w - 28 = 0
Now we can use factoring or the quadratic formula to solve for w.
After factoring or using the quadratic formula, we find that w = 4 or w = -7/2. Since width cannot be negative, we can discard the negative value.
So, the width of the rectangle is 4ft.
Now we can use this value to find the length:
l = 2w + 1
l = 2(4) + 1
l = 8 + 1
l = 9ft
Therefore, the dimensions of the rectangle are:
l = 9ft
w = 4ft
To find the dimensions of the rectangle, we need to solve the problem step by step.
Let's assume the width of the rectangle is denoted by "w" ft.
According to the problem, the length of the rectangle is 1 ft more than twice its width. This can be expressed as:
Length (l) = 2w + 1
We also know that the area of the rectangle is given by:
Area (A) = Length × Width
Given that the area of the rectangle is 28 ft², we can write:
28 = (2w + 1) × w
Now, let's solve the equation to find the value of "w":
28 = 2w² + w
Rearranging the equation:
2w² + w - 28 = 0
This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring is not possible in this case, so let's use the quadratic formula:
w = (-b ± √(b² - 4ac)) / (2a)
In our case:
a = 2, b = 1, c = -28
Substituting these values into the quadratic formula:
w = (-1 ± √(1² - 4(2)(-28))) / (2(2))
Simplifying further:
w = (-1 ± √(1 + 224)) / 4
w = (-1 ± √225) / 4
w = (-1 ± 15) / 4
This gives two possible values for "w":
w₁ = (-1 + 15) / 4 = 14 / 4 = 3.5
w₂ = (-1 - 15) / 4 = -16 / 4 = -4
However, since the width cannot be negative, we ignore the value w₂ = -4.
Therefore, the width of the rectangle is w = 3.5 ft.
To find the length (l), we substitute the value of the width into the expression for length:
l = 2w + 1 = 2(3.5) + 1 = 7 + 1 = 8 ft
Hence, the dimensions of the rectangle are:
Length (l) = 8 ft
Width (w) = 3.5 ft