A flagpole is erected at the top of a building. The angle of elevation of the top and foot of the flagpole from a point 100 m away, on the same level as that of the foot of the building, are 33° and 31° respectively. Find the height of the flagpole. Give your answer correct to the nearest metre.
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Let height of building (BC) be h meters and height of flagpole (CD) be x meters.
Distance from point = 100 m
In triangle ABC,
$$\begin{aligned}\Rightarrow \tan 31° = \frac{BC}{AB} \\[1em] \Rightarrow \tan 31° = \frac{h}{100} \\[1em] \Rightarrow 0.6009 = \frac{h}{100} \\[1em] \Rightarrow h = 0.6009 \times 100 \\[1em] \Rightarrow h = 60.09 m\end{aligned}$$
In triangle ADB,
$$\begin{aligned}\Rightarrow \tan 33° = \frac{DB}{AB} \\[1em] \Rightarrow \tan 33° = \frac{h + x}{100} \\[1em] \Rightarrow h + x = 0.6494 \times 100 \\[1em] \Rightarrow h + x = 64.94 \\[1em] \Rightarrow 60.09 + x = 64.94 \\[1em] \Rightarrow x = 64.94 - 60.09 \\[1em] \Rightarrow x = 4.85 \text{ m} \approx 5 \text{ m}.\end{aligned}$$
Hence, the height of flagpole = 5 m.