Depression changes from 30° to 60°: From the top of a 100 m tower, the angle of depression of a car is first 30° and later 60°. Assuming the car moves in a straight line toward the tower at constant level, find the distance the car travels between.

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Solve this multiple-choice question and choose the correct option.

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Introduction / Context:Angles of depression equal angles of elevation from the same horizontal line. We compare two right triangles formed by the car’s positions and the tower height. Given Data / Assumptions: Tower height H = 100 m. First depression = 30°; second depression = 60°. Car moves horizontally toward the tower on level ground. Concept / Approach:Horizontal distance d = H / tan(depression). The travel distance equals d1 - d2 between the two instants. Step-by-Step Solution: d1 = 100 / tan 30° = 100 √3.d2 = 100 / tan 60° = 100 / √3.Travel = d1 - d2 = 100 √3 - 100/√3 = (100(3 - 1))/√3 = 200/√3 = 200 √3 / 3. Verification / Alternative check:Numeric: √3 ≈ 1.732 → 200/1.732 ≈ 115.47 m. This matches 200 √3 / 3 ≈ 115.47 m. Why Other Options Are Wrong:100 √3 and 200 √3 are too large; 100 √3 / 3 is half the correct value; only 200 √3 / 3 matches the geometry. Common Pitfalls:Using cot instead of tan (or forgetting that depression equals elevation), subtracting in wrong order, or mixing degrees and radians in calculators. Final Answer:200 √3 / 3

Depression changes from 30° to 60°: From the top of a 100 m tower, the angle of depression of a car is first 30° and later 60°. Assuming the car moves in a straight line toward the tower at constant level, find the distance the car travels between the two observations.

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