Question

sᴏʟᴠᴇ ᴘʟᴇᴀsᴇ :

The elevation of the summit of a mountain from its foot is 45°. After
ascending 1 km towards the mountain upon an incline of 30°, the elevation
changes to 60°. The height of the mountain is?​

Answer 1

Answer:

Step-by-step explanation:

Let A be the foot and C be the summit of a mountain.

Given that ∠CAB = 45°

From the diagram, CB is the height of the mountain. Let CB = x

Let D be the point after ascending 2 km towards the mountain such that AD = 2 km and given that ∠DAY = 30°

It is also given that from the point D, the elevation is 60°

i.e., ∠CDE = 60°

FromtherightΔABC,tan45∘=CBAB

⇒1=xAB

  [∵ CB = x(the height of the mountain)]

⇒AB=x......(1)FromtherightΔAYD,sin30∘=DYAD

⇒12=DY2

  (∵ Given that AD = 2)

⇒DY=1.......(2)cos30∘=AYAD

⇒3–√2=AY2

  (∵ Given that AD = 2)

⇒AY=3–√......(3)FromtherightΔCED,tan60∘=CEDE

⇒tan60∘=(CB−EB)YB

   [∵ CE = (CB - EB) and DE = YB)]

⇒tan60∘=(CB−DY)AB−AY

   [∵ EB = DY and YB = (AB - AY)]

⇒tan60∘=(x−1)(x−3–√)

[∵ CB = x, DY = 1(eq:2), AB = x(eq:1) and AY = 3–√

(eq:3)]

⇒3–√=(x−1)(x−3–√)⇒x3–√−3=x−1⇒x(3–√−1)=2⇒0.73x=2⇒x=20.73=2.7

i.e., the height of the mountain = 2.7 km