Multiple choice

A tree is broken by the wind. Its top struck the ground at an angle ${30}^{o}$ at a distance of $30m$ from its foot. The whole height of the tree is:

  1. $10\sqrt 3m$
  2. $20\sqrt 3m$
  3. $40\sqrt 3m$
  4. $30\sqrt 3m$
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D Correct answer
Explanation

Let the tree height be H = x + y, where x is the standing part and y is the broken part. tan(30) = x/30 => x = 30 * (1/sqrt(3)) = 10*sqrt(3). cos(30) = 30/y => y = 30 / (sqrt(3)/2) = 60/sqrt(3) = 20*sqrt(3). Total height = 10*sqrt(3) + 20*sqrt(3) = 30*sqrt(3).

AI explanation

Let the unbroken part of the tree be y and the broken part that touches the ground be x. The broken part forms a 30 degree angle with the ground, and the distance from the foot to the top is 30 m. Using the cosine ratio, cos 30 degrees equals 30 divided by x, so x equals 30 divided by root 3 divided by 2, which simplifies to 60 divided by root 3 or 20 root 3 m. Using the sine ratio, sin 30 degrees equals y divided by x, giving y equal to half of x, which is 10 root 3 m. The total height of the tree is x plus y, so 20 root 3 plus 10 root 3 equals 30 root 3 m.

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