Discussion
From a window A, 10 m above the ground, the angle of elevation of the top C of a tower is \( x^{\circ} \), where \( \tan x^{\circ} = \dfrac{5}{2} \), and the angle of depression of the foot D of the tower is \( y^{\circ} \), where \( \tan y^{\circ} = \dfrac{1}{4} \). Calculate the height CD of the tower.
Answer: C
Work out the horizontal distance first, using the depression of the tower's foot: the window is 10 m above ground, so that 10 m is the opposite side.
\[ \tan y^{\circ} = \frac{10}{d} = \frac{1}{4} \quad\Longrightarrow\quad d = 40 \text{ m} \]
Now the elevation of the tower's top gives the part above window level:
\[ \tan x^{\circ} = \frac{h}{40} = \frac{5}{2} \quad\Longrightarrow\quad h = 100 \text{ m} \]
The full tower runs from the ground to C, so add the window's own height:
\[ CD = 100 + 10 = 110 \text{ m} \]
Forgetting the 10 m leaves 100 m, which is why that value is not offered - the near miss on the list is 70 m, from using \( \tan y \) the wrong way up.
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