Question

A kite is flying at a height of \(60 \text{ m}\) above the ground level. Ravi, standing at the roof of the house is holding the string straight and observes the angle of elevation of kite as \(30^{\circ}\). From the bottom of the same building, the angle of elevation of kite is \(45^{\circ}\). Find the length of the string and height of roof from the ground. (Use \(\sqrt{3} = 1.73\))

Hint:

In height and distance problems, always look for the shared horizontal distance between two observations. If one angle is \(45^{\circ}\), the height and distance are equal, which simplifies the calculations for the second triangle significantly.

Answer 1

Step 1: Finding Horizontal Distance:
Height of kite = 60 m
Angle of elevation from ground = 45°

Using tan 45°:
tan 45° = Opposite / Adjacent
1 = 60 / x
x = 60 m

Step 2: Finding Height of Roof:
Let height of roof = h
Angle of elevation from roof = 30°

Height difference between kite and roof = 60 − h

Using tan 30°:
tan 30° = (60 − h) / 60
1/√3 = (60 − h) / 60

60 − h = 60/√3
60 − h = 20√3
h = 60 − 20√3

Taking √3 ≈ 1.73
h = 60 − 34.6
h ≈ 25.4 m

Step 3: Finding Length of String:
Let string length = s

Using sin 30°:
sin 30° = (60 − h) / s
1/2 = (20√3) / s

s = 40√3
s ≈ 40 × 1.73
s ≈ 69.2 m

Final Answer:
Height of roof = 25.4 m
Length of string = 69.2 m