Question

Climate change and global warming are influencing storm behaviour, particularly in terms of intensity and rainfall. Strong winds and storms often cause uprooting and/or breaking of trees, which damage the vehicles standing underneath the trees. On a particular day, during a high intensity storm, a tree broke such that its broken part formed an angle of 30° with the ground. The distance between the base of the tree to the point where the top touches the ground is found to be 10 m.

Based on the above information, answer the following questions:

(i) Represent the given information with the help of a neat and well-labelled diagram.   [1]

(ii) Find the height above the ground at which the tree is broken.   [1]

(iii) (a) Find the height of the tree before it broke. (Use `sqrt(3) = 1.732`)   [2]

OR

(iii) (b) If another tree broke from the same height as in part (ii), but the broken part made a 60° angle with the ground, find the total height of the tree.   [2]

Answer 1

(i) Labelled diagram:

Right-Angled Triangle ABC Representing a Broken Tree

(ii) Height at which tree is broken (AB):

In △ABC, ∠B = 90° and ∠C = 30°

BC = 10 m

`tan 30^circ = (AB)/(BC)`

`1/sqrt(3) = (AB)/10`

`AB = 10/sqrt(3) m`

(iii) (a) Total height of the tree:

Total height = AB + AC

In △ABC:

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

`sqrt(3)/2 = 10/(AC)`

`AC = 20/sqrt(3) m`

Total height = `10/sqrt(3) + 20/sqrt(3) = 30/sqrt(3)`

= `10sqrt(3) m`

= 10 × 1.732

= 17.32 m

OR

(iii) (b) Total height of new tree:

Given: Height of broken part above ground `(AB) = 10/sqrt(3) m`

New angle with ground = 60°

Let the new broken part be AC"

In △ABC':

`sin 60^circ = (AB)/(AC'')`

`sqrt(3)/2 = (10sqrt(3))/(AC'')`

`AC'' = 10/sqrt(3) xx 2/sqrt(3) = 20/3 m`

Total height = AB + AC''

= `10/sqrt(3) + 20/3`

= `(10sqrt(3))/3 + 20/3`

= `(10(1.732) + 20)/3`

= `(17.32 + 20)/3`

= `(37.32)/3`

= 12.44 m