I need some help.
A parcel of land is 6 ft longer than it is wide. The diagonal from one corner to the opposite corner is 174ft. What are the dimensions of the parcel?
Thanks for your help.
Let the short side length be x so that the long side length will be x + 6. Then use the Pythagorean theorem:
x^2 + (x+6)^2 = 174^2
2x^2 + 12 x + 36 = 30,276
x^2 + 6 x - 15,120 = 0
That can be factored to
(x + 126)(x - 120) = 0
Take it from there
Hey there! Sure, I'd be happy to help you with a parcel of land that seems to be playing with its dimensions!
Let's assume the width of the parcel is W ft. According to the given information, the length is 6 ft longer than the width, so the length would be W + 6 ft.
Now, since we have a right-angled triangle formed by the diagonal, we can apply a little Pythagorean magic! The diagonal of 174 ft acts as the hypotenuse, and the width and length are the other two sides.
So, using Pythagoras' theorem (a^2 + b^2 = c^2) where a and b are the width and length respectively, we have:
W^2 + (W + 6)^2 = 174^2
Now, it's time to get rid of those square roots and solve for W. Get ready for some math gymnastics!
After solving, you'll find that the width of the parcel is approximately 61 ft, and the length would be 67 ft (61 + 6 = 67 ft).
So, the dimensions of the parcel are approximately 61 ft by 67 ft.
Now that we've solved that math puzzle, I hope your land isn't too much of a clown! Let me know if there's anything else I can help you with!
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.
To find the dimensions of the parcel, we can set up a system of equations based on the given information.
Let's assume the width of the parcel is "w" feet. According to the problem, the length is 6 feet longer than the width, so the length will be "w + 6" feet.
Now, we can apply the Pythagorean theorem to find the length of the diagonal:
(diagonal)^2 = (length)^2 + (width)^2
Substituting the given values into the equation, we get:
(174)^2 = (w + 6)^2 + w^2
Simplifying the equation, we have:
(174)^2 = w^2 + 12w + 36 + w^2
Expanding and rearranging the equation:
0 = 2w^2 + 12w + 36 - (174)^2
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to solve for "w":
w = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, the values of a, b, and c are:
a = 2
b = 12
c = 36 - (174)^2
Now, plug these values into the quadratic formula and calculate "w".
Once you find the value of "w", you can substitute it back into the equations:
Length = w + 6
Width = w
These will give you the dimensions of the parcel.