[tex] \mathfrak {\text{MATHS - CLASS X}} [/tex]
[tex] \mathcal{\text{SOME APPLICATIONS OF TRIGONOMETRY}} [/tex]
The angles of depression of the top and the bottom of an 8m tall building from the top of a multi-storey building are 30° and 45° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.
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[tex] \mathcal{\text{SOME APPLICATIONS OF TRIGONOMETRY}} [/tex]
The angles of depression of the top and the bottom of an 8m tall building from the top of a multi-storey building are 30° and 45° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.
[tex] \mathbb{\text{NO SPAMMING}} [/tex]
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Verified answer
The angles of depression of the top and the bottom of an 8m tall building from the top of a multi-storey building are 30° and 45° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.I have created a diagram for this case.
Here, Red line indicates the building.
Orange line indicates the multi storeyed building.
SOLUTION :
Given that, Angle of depression of top of build = α = 30°
Angle of depression of bottom of the building = β = 45°
And height of the building = 8m
Now,
DF || EA,
So Angle DAE = FDA = 30°
Also, DF || BC
Angle FDB = DBC = 45°
Now,
AB = EC = 8m .
Let the height of the multi storeyed building = x
and distance between the two is p.
So ,
DE = x - 8 .
In ΔDBC,
tan45 = DC / CB
x / p = 1
x = p --------> equation 1 .
In ΔDAE,
tan30 = DE/ AE
1/√3 = x - 8 / p
p = √3 ( x - 8 )
x = √3x - 8√3
8√3 = √3x - x
Substituting √3 = 1.73 we get
x = 8( 1.73 ) / 1.73 - 1
x = 13.84 / 0.73
x = 18.95m approximately.
Therefore, The height of the multistoreyed building is 18.95m and distance between the two buildings is also 18.95m .
HOPE THIS HELPS!
Verified answer
The angles of depression of the top and the bottom of an 8m tall building from the top of a multi-storey building are 30° and 45° respectively.
The height of multistorey building = " h " m
The distance between multistorey building and small building = " x " m
Consider all the statement from the diagram in above statement
According to diagram :
The height of multi - storey building = AB = " h " m
The height of small building = CD = 8 m
DC = EB = 8 m
AB = AE + EB
h = AE + 8
AE = ( h - 8 ) m
Now In ∆ DAE
we get equation (1)
In ∆ ABC
we have DE = CB
we get equation (2)
Now equate the equation 1 & 2
h = √3( h - 8 ) m
h = √3h - 8√3
√3h - h = 8√3
h( √3 - 1 ) = 8√3
Rationalising the denominator and numerator
AB = 4( 3 + 1.732)
AB = 4(4.732)
AB = 18.928 m
The height of multistorey building is 18.928 m
and according to equation second x = h = 18.928 m