Multiple choice

A tree is broken at certain height and its upper part $9\sqrt2m$ long not completely separated meet the ground at an angle of ${45}^{o}$. Find the height of the tree before it was broken and also find the distance from the root of the tree to the point where the top of the tree meets the ground.

  1. $8\sqrt 2$; $11m$
  2. $7(\sqrt 2-1)$; $15m$
  3. $9(\sqrt 2+1)$; $9m$

  4. None of these
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C Correct answer
Explanation

Let x be the height of the broken part. The tree forms a right triangle with the ground. sin(45) = height/x => height = x*sin(45) = 9*sqrt(2) * (1/sqrt(2)) = 9. The base distance = x*cos(45) = 9. Total height = 9 + 9*sqrt(2) = 9(1+sqrt(2)).

AI explanation

The upper part of the tree forms a right triangle with the ground, making an angle of 45 degrees. Using basic trigonometry, the distance from the root to the top is 9 sqrt(2) cos(45), which equals 9 meters, and the standing stump height is 9 sqrt(2) sin(45), which also equals 9 meters. The original height of the tree is the sum of the stump and the broken part, 9 + 9 sqrt(2), which factors to 9(sqrt(2) + 1). The height is 9(sqrt(2) + 1) and the distance is 9 meters.

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