Discussion
The angles of depression and elevation of the top of a wall 11 m high from top and bottom of a tree are 60° and 30° respectively. What is the height of the tree?
Answer: D
The tree is taller than the wall, which is why the wall's top is seen below the horizontal from the tree top.
From the tree's bottom, the wall top is at 30°:
\[ \tan 30^{\circ} = \frac{11}{d} \quad\Longrightarrow\quad d = 11\sqrt{3} = 19.05 \text{ m} \]
From the tree's top, the wall top is \( H - 11 \) below, at 60°:
\[ \tan 60^{\circ} = \frac{H - 11}{11\sqrt{3}} \quad\Longrightarrow\quad H - 11 = \sqrt{3} \times 11\sqrt{3} = 33 \]
\[ H = 33 + 11 = 44 \text{ m} \]
The tree turns out to be exactly four times the wall's height - a consequence of \( \tan 60^{\circ}\tan 30^{\circ} = 1 \) giving the neat factor of 3.
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