Question

A flag pole 'h' metres is on the top of hemispherical dome of radius "r" meters. A man is standing 7m away from the dome. Seeing the top of the pole at an angle 45° and moving 5m away from the dome and seeing the bottom of the pole at an angle 30°. Find (i) the height of the pole (ii) radius of the dome.
$( \sqrt{3} = 1.732)$
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Answer 1

Height of the pole AB = 7m

BC = CD = rm

DE = 7m

EF = 5m

from ∆ACE:

tan 45° = AC/CE

$1 = \frac{h + r}{7 + r}$

$7 + r = h + r$

$h = 7m$

from ∆BCF:

tan 30° = BC/CF

$\frac{1}{ \sqrt{3} } = \frac{r}{r + 7 + 5}$

$12 + 4 = r \sqrt{3}$

$12 = r \sqrt{3} - r$

$12 = r( \sqrt{3} - 1)$

$12 = r(1.732 - 1)$

$12 = r(0.732)$

$\frac{12}{0.732} = r$

$r = 16.39$

Therefore,

Height of the dome = 7m

Radius of the dome = 16.39