Multiple choice

When a eucalyptus tree is broken by strong wind, its top strikes the ground at an angle of $30^{\circ}$ and at a distance of $15$ m from the foot. What is the height of the tree?

  1. $15$ $\sqrt{3}$
  2. $10$ $\sqrt{3}$m
  3. $20$ m
  4. $10$ m
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A Correct answer
Explanation

Let the tree height be H. The broken part forms the hypotenuse (c), and the standing part is the height (a). tan(30) = a / 15, so a = 15 * (1/sqrt(3)) = 5*sqrt(3). cos(30) = 15 / c, so c = 15 / (sqrt(3)/2) = 30/sqrt(3) = 10*sqrt(3). Total height = a + c = 5*sqrt(3) + 10*sqrt(3) = 15*sqrt(3).

AI explanation

The standing portion of the tree acts as the opposite side to the 30 degree angle, giving it a height of 15 tan(30 degrees), which equals 15 divided by the square root of 3. The broken part forms the hypotenuse and is found using the cosine ratio as 15 sec(30 degrees), which equals 30 divided by the square root of 3. The total height of the tree is the sum of the two parts, yielding 45 divided by the square root of 3, which rationalizes to 15 times the square root of 3.

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