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?

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)

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

To find the height of the hill, we can use trigonometric ratios.

Let ( h ) be the height of the hill.

From the given information, we can set up the following equations:

For the first kilometer stone: [ \tan(30^\circ) = \frac{h}{1} ]

For the second kilometer stone: [ \tan(45^\circ) = \frac{h}{2} ]

Now, we can solve these equations for ( h ):

[ h = 1 \cdot \tan(30^\circ) ] [ h = \sqrt{3} ]

[ h = 2 \cdot \tan(45^\circ) ] [ h = 2 ]

Therefore, the height of the hill is ( \sqrt{3} ) kilometers.