Question 1 :
The angles of elevation of an artificial earth satellite is measured from two earth stations, situated on the same side of the satellite, are found to be 30° and 60°. The two earth stations and the satellite are in the same vertical plane. If the distance between the earth stations is 4000 km, find the distance between the satellite and earth. (√3 = 1.732)
Solution :
BC = x m
In triangle ABC :
∠ACB = 60°
tan θ = opposite side/Adjacent side
tan 60° = AB/BC
√3 = AB/BC
AB = √3BC
AB = x√3 ----(1)
In triangle ABD :
∠ADB = 30°
tan θ = opposite side/Adjacent side
tan 30° = AB/BD
1/√3 = AB/(BC + CD)
1/√3 = AB/(x + 4000)
AB = (x + 4000)/√3 ---->(2)
(1) = (2)
x√3 = (x + 4000)/√3
3 x = x + 4000
3x - x = 4000
2x = 4000
x = 2000
distance between the satellite and earth(AB) = x√3
= 2000 (1.732) = 3464 km
Question 2 :
From the top of a tower of height 60 m, the angles of depression of the top and the bottom of a building are observed to be 30° and 60° respectively. Find the height of the building.
Solution :
From the given information, we can draw a rough diagram
In triangle ABC :
Let AB = x and BD = 60 - x
∠ACB = 30°
tan θ = opposite side/Adjacent side
tan 30° = AB/BC
1/√3 = x/BC
BC = x √3 ----(1)
In triangle ADE :
∠AED = 60°
tan θ = opposite side/Adjacent side
tan 60° = AD/DE
√3 = 60/DE
DE = 60/√3 ----(2)
(1) = (2)
x √3 = 60/√3
3x = 60
x = 20 m
CE = 60 - 20 = 40 m
Height of the building = 40 m.
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