Question

A flag pole ‘h’ metres is on the top of the hemispherical dome of radius ‘r’ metres. A man is standing 7 m away from the dome. Seeing the top of the pole at an angle 45° and moving 5 m away from the dome and seeing the bottom of the pole at an angle 30°. Find radius of the dome `(sqrt(3) = 1.732)`

Answer 1

Height of the Flag pole (ED) = h m

AF and AD is the radius of the semi circle (r)

AC = (r + 7)

AB = (r + 7 + 5)

= (r + 12)

In the right ∆ABD, tan 30° = `"AD"/"AB"`

`1/sqrt(3) = r/("r" + 12)`

`sqrt(3` r = r + 12

`sqrt(3)` r − r = 12

⇒ `"r" (sqrt(3) - 1)` = 12

r[1.732 – 1] = 12

⇒ 0.732r = 12

r = `12/(0.732)` ⇒ = 16.39 m

In the right ∆ACE, tan 45° = `"AE"/"AC"`

`1 + ("r" + "h")/("r" + 7)`

r + 7 = r + h

∴ h = 7 m

Radius of the dome (r) = 16.39 m