Question

A person observes the angle of elevation of the peak of a hill from a station to be α. He walks c meters along a slope inclined at the angle β and finds the angle of elevation of the peak of the hill to be γ. Show that the height of the peal above the ground is

Answer 1

Let AB be the peak of a hill, so the at the station A the elevation of its summit is α°

So, ∠CAB = α°

Now when moving on the slope of β° by a distance of ‘c’m,

i.e., AD is the distance moved on the slope of β° towards the hill,

Hence AD = ’c’m………..(i)

And ∠DAF = β°

Let EB = FD = x……..(ii)

DE = FB = z…….(iii)

AF = y and CE = t………(iv)

So after moving ‘c’m, the elevation becomes γ°,

So ∠CDE = γ°

In ΔDFA

⇒ tan α (c cos β + t cot γ ) = t + c sin β

⇒ (c tan α cos β + t tan α cot γ ) = t + c sin β

⇒ t - t tan α cot γ = c tan α cos β - c sin β

⇒ t (1 - tan α cot γ ) = c(tan α cos β - sin β)

Now, AB = AE + EB = t + x

So the height of the hill is

Hence proved