What is the height of a mountain when the distance between its angles of elevation which are 16.6 degrees and 22.3 degrees on level ground is 500 meters?

Answer 1

This can be solved using the Tangent trigonometric ratio.

The larger angle of elevation is the closer angle.

Let the horizontal distance from this point to the peak of the mountain be d metres; then:

height = d × tan 22.3°

The second point is a further 500 m from the peak with an elevation of 16.6°; this gives:

height = (500 + d) × tan 16.6°

As the height is the same at both measurements, these two values must be equal:

→ d × tan 22.3° = (500 + d) × tan 16.6°

→ d × (tan 22.3° - tan 16.6°) = 500 × tan 16.6°

→ d = 500 × tan 16.6° / (tan 22.3° - tan 16.6°)

→ height = d × tan 22.3°

= 500 × tan 16.6° / (tan 22.3° - tan 16.6°) × tan 22.3°

≈ 546 m

(Generally mountains are more than 600 m, so this is more of a hill.)