One observer estimates the angle of elevation to the basket of a hot air balloon to be $60°$, while another observer $100$ m away estimates the angle of elevation to be $30°$. Find:
(a) The height of

Question

One observer estimates the angle of elevation to the basket of a hot air balloon to be $60°$, while another observer $100$ m away estimates the angle of elevation to be $30°$. Find:
(a) The height of the basket from the ground.
(b) The distance of the basket from the first observer's eye.
(c) The horizontal distance of the second observer from the basket.

Accepted Answer

Step 1: Hot air balloon with two observers, angles of elevation 60° and 30°, 100 m apart
Step 2: Let $B$ be the basket. $	an 60° = \sqrt{3} = \frac{h}{x} \Rightarrow h = x\sqrt{3}$ ... (i)
Step 3: $	an 30° = \frac{1}{\sqrt{3}} = \frac{h}{x + 100} \Rightarrow x = h\sqrt{3} - 100$ ... (ii)
Step 4: (a) Using (i) and (ii): $h = (h\sqrt{3} - 100)\sqrt{3} = 3h - 100\sqrt{3} \Rightarrow h = 50\sqrt{3}$ m
Step 5: (b) $\sin 60° = \frac{\sqrt{3}}{2} = \frac{h}{y} \Rightarrow y = \frac{50\sqrt{3}}{\sqrt{3}/2} = 100$ m
Step 6: (c) $x = \frac{h}{\sqrt{3}} = 50$ m $\Rightarrow$ Horizontal distance of second observer $= x + 100 = 150$ m

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