A tree broke due to storm at a point but didn't get separate. Its top touched the ground at a distance of 10m from its base. If the height of the point from the ground,at which the tree was broken is 12/25of the total height of the tree, find its height.
use k as a proportional constant ... 25k - 12k = 13k
13k is the hypotenuse , and 12k is one side
... this is a Pythagorean triple ... 5-12-13
... 5k = 10 m
25k (the height of the tree) is 50 m
To solve this problem, we can start by assigning variables to the unknowns we need to find. Let's call the total height of the tree "h" and the height at which the tree was broken "x".
From the given information, we know that the top of the tree touched the ground at a distance of 10m from its base. This indicates that the distance from the base to the point where the tree was broken is (10 - x) meters.
We are also given that the height at which the tree was broken is 12/25 of the total height of the tree. This can be expressed as:
x = (12/25) * h
Now, we can set up a right-angled triangle to visualize the situation. The distance from the base to the point where the tree was broken forms the base of the triangle, which has a length of (10 - x) meters. The height of the triangle is x meters (the height at which the tree was broken), and the hypotenuse represents the total height of the tree, which is h meters.
By the Pythagorean theorem, we know that the square of the hypotenuse (h^2) is equal to the sum of the squares of the other two sides. Using this, we can write the equation:
(h^2) = (10 - x)^2 + x^2
Simplifying this equation, we have:
h^2 = (100 - 20x + x^2) + x^2
h^2 = 100 - 20x + 2x^2
Now, substitute the value of x from the previous equation:
h^2 = 100 - 20((12/25) * h) + 2((12/25) * h)^2
Expanding and simplifying further, we get:
h^2 = 100 - (24/5)h + (144/625)h^2
Bring all terms to one side of the equation:
(144/625)h^2 + (24/5)h - 100 = 0
This is a quadratic equation in terms of h. To find the height of the tree, we can solve this equation using factorization, completing the square, or quadratic formula.
Once you have the solutions for h, remember that the height of a tree cannot be negative, so discard any negative solutions. The positive solution will give you the height of the tree.
I hope this explanation helps you understand how to approach and solve the problem.