\( \alpha + \beta = 90\) 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.