A man observes the angle of elevation of the top of mountain to be 30°. He walks 1000 feet nearer and finds the angle of elevation to be 45°. What is the distance of the first point of observation from the foot of the mountain?
\[ \tan 30^{\circ} = \frac{x}{x + 1000} \] \[ \frac{1}{\sqrt{3}} = \frac{x}{x + 1000} \Rightarrow x + 1000 = \sqrt{3}x \] \[ \Rightarrow (\sqrt{3} - 1)x = 1000 \Rightarrow x = \frac{1000}{\sqrt{3} - 1} \] The distance of the first point of observation from the foot of the mountain: \[ = \frac{1000}{\sqrt{3} - 1} + 1000 = 500\sqrt{3}(\sqrt{3} + 1) \, \text{ft} \]
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