Question

A tree breaks due to the storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance from the foot of the tree to the point where the top touches the ground is 10 metres. Find the height of the tree.

Answer 1

The height of the tree has to be calculated. We have to use the Trigonometric ratio to calculate the exact height of the tree.

Explanation:

We have the following set of information given to us:-

Distance between the foot of the tree and the point where the tip touches the ground = 10 m.

Angle made between the tip of the tree and the ground =30˚.

So, now if we look at the figure provided,

We have angle and adjacent side given.

We have to find the opposite side (which is the height of the broken part).

The total height of the tree will be = height of the broken part + the hypotenuse part

First we find the height of the broken part.

Tan ∅ = (op.side)/(ad.side)     =   (op.side)/(10 m)

But tan 30 = 0.57

So, 0.57 = (op.side)/10

Or, op. side = 0.57 x 10 m

Op. side = 5.7 m.

Now the height of the broken part is 5.7 m.

Total height of the tree will be calculated using Pythagorean theorem.

AB^2 + BC^2 = AC^2

AC^2 = 5.7^2 + 10^2

AC^2 = 32.49 + 100

AC^2 = 132.49

AC = √132.49

AC = 11.51m

Now the total height of the tree = standing part + broken part

=5.7 + 11.51  

= 17.21m (app).

The height of the tree will be 17.21 m (when not broken).  

(figure attached)

Answer 2

Let length of tree before windstorm is BD.

After windstorm the upper part of tree C falls from point C to point A on the ground

. Now, let CD = AC = h2 m AB = 10 m

Broken part makes an angle 60° from the ground.

So, ∠CAB = 60°

From right angled ∆ABC tan 60° = BC/AB ⇒ √3 = h1/10 ⇒ h1 = 10√3 and cos 60° = AB/AC ⇒ 1/2 = 10/h2 ⇒ h2 = 10 × 2 = 20 m

Hence, total length of tree BD = BC + CD = h1 + h1

= 10√3 + 20

= 10 × 1.732 + 20

= 17.32 + 20

= 37.32 m

Hence, height of the tree 37.32 m.