Clear height of flood light tower (stadium)

Let O be the instrument station and $D$ the horizontal distance from O to the tower. The staff is held at the plinth (foot) of the tower; the readings 5.00 m and 2.00 m are heights above the plinth. The instrument axis is at $1260.00 + 1.42 = 1261.42$ m.

                    * top (45° elevation)
                  / |
                /   |
              /     |  tower
   axis O --/---- * 5.00 m   (2°30')
              \--- * 2.00 m   (1°15')
                    |_ plinth
   |<------ D ----->|

Elevation angles

• To reading 5.00 m: $\alpha_1 = 90^\circ - 87^\circ30' = 2^\circ30'$
• To reading 2.00 m: $\alpha_2 = 90^\circ - 88^\circ45' = 1^\circ15'$
• To top: $\alpha_3 = 90^\circ - 45^\circ = 45^\circ$

Horizontal distance

Both lines of sight meet the vertical line of the tower, so

$$D\tan\alpha_1 - D\tan\alpha_2 = 5.00 - 2.00$$ $$D = \frac{3.00}{\tan 2^\circ30' - \tan 1^\circ15'} = \frac{3.00}{0.04366 - 0.02182} = 137.357\ \text{m}$$

RL of the plinth and the top

• Plinth: $RL = 1261.42 + 137.357\tan 2^\circ30' - 5.00 = 1262.417$ m
• Top: $RL = 1261.42 + 137.357\tan 45^\circ = 1398.777$ m

Clear height

$$H = 1398.777 - 1262.417 = 136.360\ \text{m}$$

Answer: clear height of the tower = 136.36 m.