Question

A surveyor is trying to find the height of a hill. He/she takes a ‘sight’ on the top of the hill and find that the angle of elevation is 40°. He/she move a distance of 150 metres on level ground directly away from the hill and takes a second ‘sight’. From this point, the angle of elevation is 22°. Find the height of the hill, correct to 1 d.p.

Answer 1

The height of the hill is 116.9 meters.

Explanation:

The diagram depicting this problem is drawn and attached below.

From Triangle ABC

`\tan 22^\circ=(h)/(150+x)\\\\h=\tan 22^\circ(150+x)`

From Triangle XBC

`\tan 40^\circ =(h)/(x)\\\\h=x\tan 40^\circ`

Therefore:

`h=\tan 22^\circ(150+x)=x\tan 40^\circ\\150\tan 22^\circ+x\tan 22^\circ=x\tan 40^\circ\\x\tan 40^\circ-x\tan 22^\circ=150\tan 22^\circ\\x(\tan 40^\circ-\tan 22^\circ)=150\tan 22^\circ\\x=(150\tan 22^\circ)/(\tan 40^\circ-\tan 22^\circ) \\\\x=139.30`

Therefore, the height of the hill

`h=139.3* \tan 40^\circ\\=116.9$ meters( correct to 1 d.p.)`

The height of the hill is 116.9 meters.