Question

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 6 m. Find the height of the tree before broken.

Answer 1

Given: A tree breaks and the top touches the ground making an angle of 30° with the ground. The horizontal distance from the foot of the tree to the touching point is AB = 6 m. Let the original tree height = h. Let BC be the remaining vertical part (height x) and CD the broken part (length h – x) which now lies along AC, so AC = h – x.

Step-wise calculation:

1. In right triangle ABC. 

Angle at A = 30°.

So, `tan 30^circ = (BC)/(AB)`.

= `x/6` 

Hence, `x = 6 xx 1/sqrt(3)`

= `6/sqrt(3)`

= `2sqrt(3)` m

2. AC = Length of broken part.

Using cos 30° in triangle ACB:

`cos 30^circ = (AB)/(AC)`

⇒ `sqrt(3)/2 = 6/(AC)`

⇒ `AC = 6 xx 2/sqrt(3)`

= `12/sqrt(3)`

= `4sqrt(3)` m

3. Original height h = BC + CD

= BC + AC

= `2sqrt(3) + 4sqrt(3)` 

= `6sqrt(3)` m

Height of the tree before breaking = `6sqrt(3)` m ≈ 10.39 m.