A rectangle has a length of 34
feet less than 9
times its width. If the area of the rectangle is 6583
square feet, find the length of the rectangle.
Let's assume the width of the rectangle is x feet.
According to the given information, the length of the rectangle is 9x - 34 feet.
The area of a rectangle can be calculated by multiplying its length by its width, so we have the equation:
(x)(9x - 34) = 6583
By distributing, we get:
9x^2 - 34x = 6583
To solve this quadratic equation, let's move 6583 to the left side:
9x^2 - 34x - 6583 = 0
Now, we can attempt to factorize.
However, this quadratic equation does not readily factorize, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 9, b = -34, and c = -6583.
Substituting the values:
x = (-(-34) ± √((-34)^2 - 4(9)(-6583))) / (2(9))
Simplifying:
x = (34 ± √(1156 + 237384)) / 18
x = (34 ± √238540) / 18
x = (34 ± 488.4) / 18
Now, let's solve for each ± separately:
1. x = (34 + 488.4) / 18
x = 522.4 / 18
x ≈ 29
2. x = (34 - 488.4) / 18
x = (-454.4) / 18
x ≈ -25.2
Since we are dealing with the measurement of length, the width cannot be negative, so we discard x ≈ -25.2.
Therefore, the width of the rectangle is approximately 29 feet.
And, the length of the rectangle is given by the equation: 9x - 34, where x = 29:
Length = 9(29) - 34
Length = 261 - 34
Length = 227 feet
Therefore, the length of the rectangle is 227 feet.