Question

From the top of the hill the angles of depression of two consecutive kilometre stones due east are found to be 45° and 30° respectively. Find the height of the hill.

Answer 1

Let the height of hill be H
Distance of the 2 stones from bose of hill be R, R+1 km since they are consecutive kilometer stones
R = H/tan45
R +1 = H/tan30
By solving
H = 1/(√3-1) km

Answer 2

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√3=ABBC+CD [tan 30 = 1√3 =1/1.732] 
or, 11.732=ABAB+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