Question

From the top of a tower 63.2 ft high, the angles of depression of two objects situated in
the same horizontal line with the base of the tower, and on the same side of the tower, are
31 0 16’ and 46 0 28’ respectively. Find the distance between the two objects.

Answer 1

\[\tan 31^\circ16'=\frac{ AB }{ BC }\]
\[AB=BC \tan 31^\circ16'\]
\[\frac{ AB }{ BD }=\tan 46^\circ 28'\]
\[AB=BD \tan 46^\circ 28.\]
\[BC \tan 31^\circ 16'=BD\tan 46^\circ 28'\]
\[(BD+DC)\tan 31^\circ16'=BD \tan 46^\circ28'\]
\[DC \tan 31^\circ16'=BD(\tan 46^\circ 28'-\tan 31^\circ16')\]

Answer 2

\[DC=\frac{ AB(\tan46^\circ28'-\tan 31^\circ16') }{( \tan 46^\circ28')(\tan 31^\circ 16') }\]
\[=63.2\frac{ \tan 46^\circ 28'-\tan 31^\circ16' }{ (\tan 46^\circ28')(\tan 31^\circ16') }\]
\[\approx 44.04 ~ft\]