The angle of elevation of the top of a tower from a point on the ground is 45°. Moving 24 m directly towards the base of the tower, the angle of elevation changes to 60°. What is the height of the tower? (Use \(\sqrt{3} = 1.732\), and round off your answer to two decimal places.)
56.78 m
Let h be the tower height and x be the distance from the closer point (angle 60°) to the foot. Then h / x = tan 60° = √3 and h / (x + 24) = tan 45° = 1.
From the second equation, h = x + 24. Substituting in h = √3 x: x + 24 = √3 x, so x (√3 − 1) = 24 and x = 24 / (√3 − 1) = 12(√3 + 1).
Therefore h = x + 24 = 12√3 + 12 + 24 = 12√3 + 36 = 12 × 1.732 + 36 = 20.784 + 36 = 56.784 ≈ 56.78 m.
Hence, the answer is 56.78 m.
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