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One tower, two points on opposite sides with given angles: find the gap or the height

This type of question, asked 2 times: 29 Jan 2022 Tier II, 2 Mar 2023 Tier II. All 2 questions below, oldest first.

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  1. Question 1

    29 Jan 2022 Tier II

    89
    The angle of elevation of the top of a tower 25325\sqrt{3} m high from two points on the level ground on its opposite sides are 45∘45^\circ and 60∘60^\circ. What is the distance (in m) between the two points (correct to one decimal place)?

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  2. Question 2

    2 Mar 2023 Tier II

    27
    Two ships are on the opposite of a light house such that all three of them are collinear. The angles of depression of the two ships from the top of the light house are 30∘30^\circ and 60∘60^\circ. If the ships are 2303230\sqrt{3} m apart, then find the height of the light house (in m).

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