Question

From the top of a hill, the angles of depression of two consecutive kilometer stones due to east are found to be 30° and 45°. Find the height of the hill (or) what is the height of the hill?

Answer 1

1.366 km

Explanation:

Let AB is the height of the hill and two stones are C and D respectively where depression is 45 degree and 30 degree. The distance between C and D is 1 km.

Here depression and hill has formed right angle triangles with the base. We have to find the height of the hill with this through trigonometry.

In triangle ABC, tan 45 = height/base = AB/BC
or, 1 = AB/BC [ As tan 45 degree = 1]
or, AB = BC ..........(i)

Again, triangle ABD, tan 30 = AB/BD
or, `1/sqrt 3 = (AB)/(BC+CD)` [tan 30 = `1/sqrt3` =1/1.732]
or, `1/1.732 = (AB)/(AB+1)` [ As AB = BC from (i) above]

or, 1.732 AB = AB +1

or, 1.732 AB - AB = 1

or, AB(1.732-1) = 1

or, AB * 0.732 = 1

or AB = 1/0.732 = 1.366

Hence height of the hill 1.366 km