A Man observes the following conditions in a
mountain:
At the foot of a mountain, the elevation of its summit
is 45°, after ascending 2 km towards the mountain
up an incline of 30°, the elevation changes to 60°
The height of the mountain is
Answers
Answer:
2.732 km
Step-by-step explanation:
Let A be the foot and B be the summit of the mountain AOB
In right-angled triangle △AOB
∠BAO=45∘
∠ABO=45∘
⇒AO=OB=h......tan45=OBOA=1(say)
In right-angled triangle △ACP
sin30∘=PCAP
⇒12=PC2
⇒PC=1 →DO=1
Also, cos30∘=ACAP
⇒3√2=AC2
⇒AC=3√
Now, h=OB=OA=OC+CA
⇒h=OC+3√
⇒OC=h−3√
⇒PD=h−3√
In right angled triangle △PDB,
tan60∘=BDPD
⇒3√=BDh−3√
⇒3√=h−1h−3√
⇒3√h−3=h−1
⇒h=23√−1=23√−1×3√+13√+1=3√+1=1.732+1=2.732
Hence height of the mountain is 2.732km
Given : At the foot of a mountain, the elevation of its summit is 45°, after ascending 2 km towards the mountain up an incline of 30°, the elevation changes to 60°
To find : The height of the mountain
Solution:
At the foot of a mountain, the elevation of its summit is 45°,
Tan 45 = Height of Summit / Distance of foot from Base of Summit
=> Height of Summit = Distance of foot from Base of Summit = H cm
ascending 2 km towards the mountain up an incline of 30°
Sin 30 = Vertical Distance / 2
=> 1/2 = Vertical Distance / 2
=>Vertical Distance Covered = 1 km
Remaining Vertical Distance = H - 1 km
Horizontal Distance Covered = √2² - 1² = √3 km
Remaining Horizontal Distance = H - √3
elevation changes to 60°
=> Tan 60 = ( H - 1)/(H - √3)
=> √3 = ( H - 1)/(H -√3)
=> H√3 - 3 = H - 1
=> H(√3 - 1) = 2
=> H = 2/(√3 - 1)
=> H = √3 + 1
=> H = 2.732
Height of Mountain = 2.732 km
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