To find the dimensions of the parcel, let's first assign a variable to represent the width of the parcel.
Let the width of the parcel be "x" ft.
According to the problem statement, the length of the parcel is 6 ft longer than its width. Therefore, the length is (x + 6) ft.
Now, using the Pythagorean theorem, we can find the length of the diagonal.
The Pythagorean theorem states that the square of the hypotenuse (diagonal) of a right triangle is equal to the sum of the squares of the other two sides.
In this case, the width (x) will be one side, and the length (x + 6) will be the other side. The diagonal is the hypotenuse.
So, the equation can be set up as:
x^2 + (x + 6)^2 = 174^2
Expanding and simplifying:
x^2 + x^2 + 12x + 36 = 30456
Combining like terms:
2x^2 + 12x + 36 = 30456
Rearranging:
2x^2 + 12x + 36 - 30456 = 0
Subtracting 30456 from both sides:
2x^2 + 12x - 30420 = 0
To solve this quadratic equation, we can make use of the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 2, b = 12, and c = -30420.
Substituting these values into the quadratic formula:
x = (-12 ± √(12^2 - 4(2)(-30420))) / (2(2))
Calculating inside the square root:
x = (-12 ± √(144 + 243840)) / 4
Simplifying inside the square root:
x = (-12 ± √243984) / 4
Now, we can find two possible values for x:
x = (-12 + √243984) / 4
x ≈ 92.7 ft
x = (-12 - √243984) / 4
x ≈ -156.7 ft (This negative value is not applicable in this context.)
So, the width of the parcel is approximately 92.7 ft.
By substituting this value back into the expression for the length (x + 6):
Length ≈ 92.7 + 6 ≈ 98.7 ft
Therefore, the dimensions of the parcel are approximately 92.7 ft by 98.7 ft.