a vertical post AB erected on a level horizontal ground and a point C located 30m away from it are shown in figure. The angle of elevation of the top of the post B , when observed from point C is 48 degrees. The length of a wire tied to B from the point D located in the same direction as C from A, is 50m. Show that the angle of elevation of B when observed from D is greater than 40 degrees.
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Step-by-step explanation:
Let y be the height of the flag post and x be the initial distance from flag post.
Now, after travelling 6 m towards the post the distance from the base of pole is (x−6) m.
We know that tanθ=
Adjacentside
Oppositeside
=
BC
AB
In △ABC, ∠B=90
0
and ∠C=30
0
.
Here, θ=30
0
, BC=x m and AB=y m, therefore,
tanθ=
BC
AB
⇒tan30
0
=
x
y
⇒
3
1
=
x
y
(∵tan30
0
=
3
1
)
⇒x=
3
y...........(1)
Also, in △ABD, ∠B=90
0
and ∠D=45
0
.
Here, θ=45
0
, BD=(x−6) m and AB=y m, therefore,
tanθ=
BD
AB
⇒tan45
0
=
x−6
y
⇒1=
x−6
y
(∵tan45
0
=1)
⇒x−6=y
⇒x=y+6...........(2)
Comparing equations 1 and 2, we get
3
y=y+6
⇒
3
y−y=6
⇒y(
3
−1)=6
⇒y=
3
−1
6
Hence, the height of the flag post is
3
−1
6
m.
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