To find the dimensions of the land, we can use the Pythagorean theorem, which relates the sides of a right triangle. In this case, the length and width form two sides of a right triangle, and the diagonal is the hypotenuse.
Let's denote the width of the land as 'w'. According to the problem, the length is 6 meters more than the width, so the length can be represented as 'w + 6'.
Using the Pythagorean theorem, we have:
(diagonal)^2 = (width)^2 + (length)^2
Replacing the variables with their values, we get:
10^2 = w^2 + (w + 6)^2
Now, we can solve this equation to find the width of the land.
Expanding and simplifying the equation, we have:
100 = w^2 + (w^2 + 12w + 36)
Combining like terms:
100 = 2w^2 + 12w + 36
Rearranging the equation to form a quadratic equation:
2w^2 + 12w + 36 - 100 = 0
2w^2 + 12w - 64 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Using the quadratic formula, we have:
w = (-b ± √(b^2 - 4ac)) / (2a)
Where a = 2, b = 12, and c = -64. Substituting these values, we get:
w = (-12 ± √(12^2 - 4(2)(-64))) / (2(2))
Simplifying the equation, we have:
w = (-12 ± √(144 + 512)) / 4
w = (-12 ± √656) / 4
w = (-12 ± 25.61) / 4
Now, we can find the two possible values for the width 'w':
w₁ = (-12 + 25.61) / 4 ≈ 3.65
w₂ = (-12 - 25.61) / 4 ≈ -9.16
Since we cannot have a negative width for the land, we conclude that the width is approximately 3.65 meters.
To find the length, we substitute this value back into the equation we used earlier:
length = w + 6
length = 3.65 + 6
length ≈ 9.65 meters
Therefore, the dimensions of the land are approximately 3.65 meters by 9.65 meters.