Question

From the top of a tower of height 50 m, the angles of depression of the top and bottom of a pillar are $45^\circ$ and $60^\circ$ respectively. Find the height of the pillar.

Hint:

Angles of depression are measured from the horizontal line of sight; their corresponding angle of elevation at the other end is equal.

Answer 1

Step 1: Let the height of the pillar be \( h \) m. 
Let the horizontal distance between the tower and pillar be \( x \) m. 
Step 2: Use trigonometric ratios for the angles of depression. 
For the top of the pillar (\(45^\circ\)): \[ \tan 45^\circ = \dfrac{50 - h}{x} \Rightarrow 1 = \dfrac{50 - h}{x} \Rightarrow x = 50 - h \] For the bottom of the pillar (\(60^\circ\)): \[ \tan 60^\circ = \dfrac{50}{x} \Rightarrow \sqrt{3} = \dfrac{50}{x} \Rightarrow x = \dfrac{50}{\sqrt{3}} \] Step 3: Equate both expressions for \(x\). 
\[ 50 - h = \dfrac{50}{\sqrt{3}} \Rightarrow h = 50 - \dfrac{50}{\sqrt{3}} \] Step 4: Simplify. 
\[ h = 50\left(1 - \dfrac{1}{\sqrt{3}}\right) = 50\left(\dfrac{\sqrt{3} - 1}{\sqrt{3}}\right) = \dfrac{50(\sqrt{3} - 1)}{1.732} \] \[ h \approx 50(0.577) = 28.85 \, \text{m} \] Step 5: Conclusion. 
Hence, the height of the pillar is approximately 28.87 m.