Heights and distances

**mathlover14** · June 9th 2011, 08:56 PM

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 !

**corsica** · June 10th 2011, 04:37 AM

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:
$a^{2}+b^{2}=c^{2}$