Discussion
A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7 m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is \( 30^{\circ} \) and that of the top of the flagstaff is \( 45^{\circ} \). Find the height of the tower.
Answer: A
Let the tower be \( h \) m and the observation point \( d \) m away. The bottom of the flagstaff is the top of the tower.
\[ \tan 30^{\circ} = \frac{h}{d} \quad\Longrightarrow\quad d = h\sqrt{3} \]
\[ \tan 45^{\circ} = \frac{h+7}{d} \quad\Longrightarrow\quad h + 7 = d = h\sqrt{3} \]
\[ h(\sqrt{3} - 1) = 7 \quad\Longrightarrow\quad h = \frac{7}{\sqrt{3}-1} = \frac{7(\sqrt{3}+1)}{2} \]
\[ h = 3.5 \times 2.732 = 9.562 \text{ m} \]
Both sightings share the same \( d \), which is what allows one to be substituted into the other.
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