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The angles of depression of top and bottom of a building 50m high as observed from
top of a tower are 30o and 60o, respectively. Find the height of the tower and also
the horizontal distance between the building and the tower.

Answer
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Hint: The given question is related to heights and distance. Try to recall Try to recall the formulae
related to trigonometric ratios and values of trigonometric functions for standard angles.
The following formulae will be used to solve the given problem:
(a) tanθ=oppositesideadjacentside
(b) tan(30o)=13
(c) tan(60o)=3

Complete step by step solution:
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Now, considering the information given in the question, we can draw the following figure for better visualization of the problem:


Let AB be the tower of height hmeter and CD be the building of height 50m . Let x
meter be the horizontal distance between the building and the tower. We will consider a point G
on the tower AB which is at the same level as the top of the building CD . So, BG=50m and
AG=(h50)m . Also, GD=x meter . In the question, it is given that the angle of
depression of top and bottom of the building as seen from the top of the tower are 30o and
60o respectively. So, , EAC=60o and EAD=30o.
Now, AE and BC are horizontal lines. So, they are parallel to each other. So, EAC and ACB are alternate interior angles. So, they will be equal. Also,
EAD and ADG are also alternate interior angles. So, they will
also be equal. So, ACB=EAC=60o and ADG=EAD=30o.
Now, we will consider ΔADG.
In ΔADG ,
tan(300)=h50x
13=h50x
x=3(h50)......(i)
Now, we will consider ΔACB .
In ΔACB,
tan(60o)=hx
3=hx
x=h3......(ii)
Now, the distance between the tower and the building is the same in both cases. So, (i)=(ii) .
3(h50)=h3

On cross-multiplying, we get:
3(h50)=h
3h150=h
2h=150
h=75m
Substituting h=75 in equation (ii) , we get:
x=753m
Hence, the height of the tower is 75m and the distance between the building and tower is equal
to 753m.

Note: Students are generally confused between the values of tan(30o) and tan(60o). tan(30o)=13 and tan(60o)=3. These values should be remembered as they are used in various problems of heights and distances.
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