49.
A man standing in one corner of a square football field observes that the angle subtend by a pole in the corner just diagonally opposite to this corner is 60°. When he retires 80 m from the corner, along the same straight line, he finds the angle to be 30°. The length of the field is
20 m
40 m
D.
Let AB = BC = x metre,
Solution:
In the figure
AB is the tower
BD and BC are the shadow of the tower in two situations
Consider BD = x m and AB = h m
In triangle ABD
tan 450 = h/x
So we get
1 = h/x
h = x ….. (1)
In triangle ABC
tan 300 = h/(x + 10)
So we get
1/√3 = h/(x + 10)
Using equation (1)
h√3 = h + 10
h (√3 – 1) = 10
We know that
h = 10/(√3 – 1)
It can be written as
h = [10 (√3 + 1)]/ [(√3 – 1) (√3 + 1)]
By further calculation
h = (10√3 + 1)/ 2
So we get
h = 5 (1.73 + 1)
h = 5 × 2.73
h = 13.65 m
Therefore, the height of the tower is 13.65 m.
When the sun's altitude changes from 30∘ to 60∘, the length of the shadow of a tower decreases by 70 m. What is the height of the tower?
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