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Math Help - Heights and distances

  1. #1
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    Heights and distances

    A straight palm tree , 60 feet high , is broken by the wind but not completely separated , and its upper part meets the ground at an angle of 30 degrees . Find the distance of the point where the tree meets the ground , from the root , and also the height at which the tree is broken .

    Please Help !
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  2. #2
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    Ok, the first thing to note is that the ground and the two parts of the tree form a right triangle (see image).

    Side a is the (still standing) lower part of the tree, side c is the (broken off) upper part of the tree, and side b is the ground.

    Since we know that the total height of the tree was 60 ft, we can write:
    a+c = 60 ft

    We now have two unknown variables, but only equation. So we need one more equation. Luckily for a right triangle we have equations that relate angles and side lenghts. The sine of an angle in the triangle equals the opposing side divided by the hypotenuse side:
    \sin 30^{\circ} = \frac{a}{c}
    By combining the two equations you can now find a and c.

    Afterwards you use Pythagoras' theorem to find the one remaining unknown side b:
    a^{2}+b^{2}=c^{2}
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  3. #3
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    it also helps to know that in a 30-60-90 triangle, c = 2a
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  4. #4
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    Quote Originally Posted by skeeter View Post
    it also helps to know that in a 30-60-90 triangle, c = 2a
    Ah, true. That's simply what you get when you write out \sin 30^{\circ}=\frac{a}{c}.
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