Let's represent the height of the tree as "h". According to the problem, the wire supporting the tree is 5 m longer than the height of the tree, so the length of the wire is "h + 5". The wire is anchored at a point whose distance from the base of the tree is 35 m shorter than the height of the tree, so this distance is "h - 35".
We can now use the Pythagorean theorem to find the height of the tree. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, we have a right triangle formed by the tree, the ground and the wire. The height of the tree is one side of this triangle, and its length is "h". The distance from the base of the tree to where the wire is anchored is another side of this triangle, and its length is "h - 35". The wire itself is the hypotenuse of this triangle, and its length is "h + 5".
Applying the Pythagorean theorem, we have:
(h + 5)² = h² + (h - 35)²
Expanding and simplifying this equation:
h² + 10h + 25 = h² + h² - 70h + 1225
2h² - 80h + 1200 = 0
Dividing by 2:
h² - 40h + 600 = 0
We can solve this quadratic equation using factoring or by applying the quadratic formula. Factoring gives us:
(h - 30)(h - 20) = 0
So, h = 30 or h = 20.
Since both solutions are positive, either one could be a valid answer for this problem. However, if we have additional information or constraints that would allow us to choose one solution over the other, we could determine which one is correct. Without such information, we can only say that there are two possible answers: either h = 30 or h = 20.