\( \alpha + \beta = 90\) and \( \delta + \beta = 90\)
\( \therefore \delta = \alpha\)
\( \tan \theta = \frac{P}{36}\) (1)
\( \tan 2 \theta = \frac{T}{36}\) (2)
\( \tan \alpha = \frac{T}{18}\) (3)
\( \tan \alpha = \frac{18}{P}\) (4)
Combine (3) and (4)
\( T = \frac{18^2}{P}\) (3.4)
Simplify (2)
\( \tan 2\theta = \frac{2 \tan \theta}{1-tan^2 \theta}\)
\( \frac{T}{36} = \frac{2 \tan \theta}{1-tan^2 \theta}\)
Combine simplified (2) and (1)
\( \frac{T}{36}= \frac{2 \frac{P}{36}}{1-(\frac{P}{36})^2}\)
\( \frac{T}{36}= \frac{72P}{36^2 - P^2}\)
\( P = \frac{2592P}{36^2-P}\) BUT \( T = \frac{18^2}{P}\)
Thus
\( \frac{18^2}{P} = \frac{2592P}{36^2-P}\)
\( P^2 = \frac{36^2-P^2}{8}\)
\( 9P^2 = 162 (8)\)
\( P = 12 m\)
\( T = \frac{18^2}{12}\)
\( T = 27 m\)
Therefore: The height of the pole is 12 meters and the height of the tower is 27 meters.