Question:hard

A kite is flying at a height of 60 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 \(\sqrt{3} = 1.73\))

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An angle of elevation of \(45^\circ\) always forms an isosceles right-angled triangle.
This immediately tells you that the horizontal ground distance is equal to the vertical height, which is 60 m!
Updated On: Jul 9, 2026
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Solution and Explanation

Step 1: Find the horizontal distance using the bottom view.
From the bottom of the building, the angle of elevation to the kite (height 60 m) is \(45^\circ\), so: \[ \tan 45^\circ = \frac{60}{x} \implies x = 60 \text{ m} \]
Step 2: Find the roof height using this same horizontal distance.
From the roof, the vertical rise to the kite is \(60-h\), and: \[ \tan 30^\circ = \frac{60-h}{60} \implies 60-h = \frac{60}{\sqrt{3}} = 20\sqrt{3} \implies h = 60 - 20(1.73) = 25.4 \text{ m} \]
Step 3: Find the string length using the Pythagoras theorem instead of the sine ratio.
The string is the straight line from the roof to the kite, forming the hypotenuse of a right triangle with legs \(x = 60\) m and \(60-h = 20\sqrt{3}\) m: \[ L^2 = 60^2 + (20\sqrt{3})^2 = 3600 + 1200 = 4800 \]
Step 4: Take the square root to get the string length.
\[ L = \sqrt{4800} = 40\sqrt{3} = 40 \times 1.73 = 69.2 \text{ m} \]
\[ \boxed{L = 69.2 \text{ m},\ h = 25.4 \text{ m}} \]
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