The area of a rectangle is 44yds squared, and the length of the rectangle is 3yds less than twice the width. what is the length and width
width ---- x
length --- 2x - 3
x(2x-3) = 44
2x^2 - 3x - 44 = 0
(x + 4)(2x - 11) = 0
x = -4, which would make no sense
or
x = 11/2 or 5.5 yds
the width is 5.5 yrds and the length is 8 yards
check:
8(5.5) = 44
a=l*w 44=3-2x , 2x=44-3, 2x=39
Let's assume the width of the rectangle is denoted by "w" yards.
According to the given information, the length of the rectangle is 3 yards less than twice the width, which can be expressed as: length = 2w - 3.
The area of a rectangle is given by the formula: area = length × width.
Substituting the given values into the formula, we have:
44 = (2w - 3) × w
Expanding the equation:
44 = 2w² - 3w
Rearranging the equation:
2w² - 3w - 44 = 0
To solve this quadratic equation, we can either factorize or use the quadratic formula. In this case, let's use the quadratic formula:
w = (-b ± √(b² - 4ac)) / 2a
For the equation 2w² - 3w - 44 = 0, a = 2, b = -3, and c = -44. Substituting these values into the formula, we get:
w = (-(-3) ± √((-3)² - 4(2)(-44))) / 2(2)
w = (3 ± √(9 + 352)) / 4
w = (3 ± √361) / 4
w = (3 ± 19) / 4
Solving for w with both signs:
w₁ = (3 + 19) / 4 = 22 / 4 = 5.5
w₂ = (3 - 19) / 4 = -16 / 4 = -4
Since width cannot be negative, we discard -4 as a non-viable solution.
Therefore, the width of the rectangle is 5.5 yards.
To find the length, substitute the width value back into the expression for length:
length = 2w - 3
length = 2(5.5) - 3
length = 11 - 3
length = 8
So, the length of the rectangle is 8 yards.
To find the length and width of the rectangle, we can use the given information and set up a system of equations.
Let's assume the width of the rectangle is 'w' yards.
According to the given information, the length of the rectangle is 3 yards less than twice the width, so it can be represented as (2w - 3) yards.
We also know that the area of a rectangle is equal to its length multiplied by its width. So, we can write an equation for the area of the rectangle:
Area = Length × Width
44 = (2w - 3) × w
To solve this quadratic equation, we can multiply it out and rearrange terms:
44 = 2w² - 3w
Bringing all the terms to one side, we have:
2w² - 3w - 44 = 0
Now, we need to solve this quadratic equation for 'w'. There are different methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula.
Let's use the quadratic formula to find the value of 'w':
w = [-b ± √(b² - 4ac)] / 2a
In our equation, a = 2, b = -3, and c = -44.
Substituting these values into the formula, we have:
w = [-(-3) ± √((-3)² - 4(2)(-44))] / (2(2))
w = [3 ± √(9 + 352)] / 4
w = [3 ± √361] / 4
w = (3 ± 19) / 4
Simplifying further, we have two possible solutions for 'w':
w₁ = (3 + 19) / 4 = 22 / 4 = 5.5
w₂ = (3 - 19) / 4 = -16 / 4 = -4
Since width cannot be negative in this context, we discard the negative value (-4).
Therefore, the width of the rectangle is 5.5 yards.
To find the length, we substitute this value back into the equation for the length:
Length = 2w - 3
Length = 2(5.5) - 3
Length = 11 - 3
Length = 8
So, the length of the rectangle is 8 yards and the width is 5.5 yards.