2018 · Q63
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
Elementary MathematicsTrigonometry
- A√2 − 1
- B(√3 + 1)/(2√2)
- C(√3 + 1)/(2√2)
- D(√3 + 1)(√2 − 1)/(2√2)
Explanation
From the second position the flag-top's elevation is 45 degrees, so the observer's distance from the dome's centre equals the flag-top height r + h. The sight line is tangent to the dome, so its distance from the centre is the radius r: (r + h)/sqrt(2) = r. This gives h = r(sqrt(2) - 1), i.e. h/r = sqrt(2) - 1. Hence option A.