Question

A flagstaff stands on a tower. At a distance 10 m from the tower, the angles of elevation of the top of the tower and flagstaff are $45^\circ$ and $60^\circ$ respectively. Find the length of the flagstaff.

Hint:

In height and distance problems, always draw a right triangle diagram and use $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ for elevation or depression.

Answer 1

Step 1: Let the height of the tower be \( h \) m and the height of the flagstaff be \( x \) m. 
Total height of tower + flagstaff = \( h + x \). Distance from the tower = \( 10 \, \text{m} \). 
Step 2: Use trigonometric ratios for each angle of elevation. 
For the top of the tower: \[ \tan 45^\circ = \dfrac{h}{10} \Rightarrow 1 = \dfrac{h}{10} \Rightarrow h = 10 \, \text{m} \] For the top of the flagstaff: \[ \tan 60^\circ = \dfrac{h + x}{10} \Rightarrow \sqrt{3} = \dfrac{h + x}{10} \Rightarrow h + x = 10\sqrt{3} \] Step 3: Substitute the value of \( h \). 
\[ 10 + x = 10\sqrt{3} \Rightarrow x = 10(\sqrt{3} - 1) \] \[ x = 10(1.732 - 1) = 7.32 \, \text{m} \] Step 4: Conclusion. 
Hence, the length of the flagstaff is approximately 7.32 m.