On the top of a hemispherical dome of radius r, there stands a flag of height h. From a point on the ground, the elevation of the top of the flag is 30°. After moving a distance d towards the dome, when the flag is just visible, the elevation is 45°. The ratio of h to r is equal to

On the top of a hemispherical dome of radius r, there stands a flag of height h. From a point on the ground, the elevation of the top of the flag is 30°. After moving a distance d towards the dome, when the flag is just visible, the elevation is 45°. The ratio of h to r is equal to Correct Answer √2 – 1

In the figure given below,

Considering ∆PQR,

⇒ tan 45° = Perpendicular/Base = PQ/QR

⇒ 1 = (h + r)/QR

⇒ QR = (h + r)

∵ The flag is just visible from the point R, PR is a tangent to the hemispherical dome

⇒ QT = Radius of dome = r

⇒ ∠QTR = ∠QTP = 90°

Considering ∆QTR,

⇒ ∠TQR = ∠QRT = 45°

This means that ∆QTR is an isosceles triangle

⇒ RT = QT = r

Similarly, considering ∆QTP,

⇒ ∠TQP = ∠QPT = 45°

This means that ∆QTP is an isosceles triangle

⇒ PT = QT = r

⇒ PR = PT + RT = 2r

Now, applying Pythagoras theorem in ∆PQR,

⇒ PR² = PQ² + QR²

⇒ (2r)² = (h + r)² + (h + r)²

⇒ 4r² = 2(h + r)²

⇒ 2r² = (h + r)²

⇒ r√2 = h + r

⇒ h = r(√2 – 1)

⇒ h/r = (√2 – 1)