Question

A tree stand svertically on a hill side which makes an angle of ${15}^{\circ }$ with the herizontal. From the same point on the ground , the slant distance of the base of the tree is 35m. Find the height of the tree, if the angle of elevation of the to of the tree is ${60}^{\circ }$.

Write the importance of tree in out life.

Answer 1

Let PQ be the tree on the hill which makes an angle of ${15}^{\circ }$ with the horizontal AR. where A is a point on the ground 35 m down the hill from the base of the tree.

In $\Delta ARQ,$We have

Also, $\angle A+\angle Q+\angle R={90}^{\circ }$

$\Rightarrow \angle Q+{90}^{\circ }+{60}^{\circ }={180}^{\circ }$

$\Rightarrow \angle Q={30}^{\circ }$

In $\Delta APQ,$We have

$\angle PAQ={45}^{\circ }$and $\angle APQ,$We get

$\frac{AP}{sin\angle AQP}=\frac{PQ}{sin\angle PAQ}$

$\Rightarrow \frac{35}{sin{30}^{\circ }}=\frac{PQ}{sin{45}^{\circ }}$

$\Rightarrow \frac{35}{1/2}=\frac{PQ}{1/\sqrt{2}}$

$\Rightarrow PQ=35\sqrt{2}m$

Hence, height of the tree is $35\sqrt{2}m.$

The importance of tree in our life are as follows:

(i) Trees contribute to the evironment by providing oxygen preserving soil and supporting wildife.

(ii) Trees control climate by moderate the effect of the sun, rain and wind.

(iii) Trees provide timber for building construction , furniture manufacture , tools and lots of household items.