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: Understanding the Concept:
This problem involves trigonometry in right-angled triangles to find heights and distances.
The horizontal distance from the building to the point directly below the kite remains constant.
Step 2: Key Formula or Approach:
Use \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\) and \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\).
Let \(H = 60 \text{ m}\) (height of kite), \(h\) be the height of the roof, and \(x\) be the horizontal distance.
Step 3: Detailed Explanation:
Let the building be \(AB\) with height \(h\). Let the kite be at point \(K\) at height \(60 \text{ m}\).
From the bottom of the building (point \(A\)):
In \(\triangle KAC\) (where \(C\) is on the ground below kite):
\[ \tan 45^{\circ} = \frac{60}{x} \Rightarrow 1 = \frac{60}{x} \Rightarrow x = 60 \text{ m} \]
From the roof (point \(B\)):
The height of the kite above the roof is \((60 - h)\).
In the right triangle formed with the roof level:
\[ \tan 30^{\circ} = \frac{60 - h}{x} \]
\[ \frac{1}{\sqrt{3}} = \frac{60 - h}{60} \]
\[ 60 - h = \frac{60}{\sqrt{3}} = 20\sqrt{3} \]
\[ h = 60 - 20(1.73) = 60 - 34.6 = 25.4 \text{ m} \]
Now, to find the length of the string (\(s\)) from Ravi:
\[ \sin 30^{\circ} = \frac{60 - h}{s} \]
\[ \frac{1}{2} = \frac{20\sqrt{3}}{s} \]
\[ s = 40\sqrt{3} = 40 \times 1.73 = 69.2 \text{ m} \]
Step 4: Final Answer:
The height of the roof is \(25.4 \text{ m}\) and the length of the string is \(69.2 \text{ m}\).