Changing depression from 30° to 45°: From the top of a vertical tower, a car approaches in a straight line on level ground. The angle of depression changes from 30° to 45° in 12 minutes. How much additional time is required (after it is 45°) for the.

Question

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

Accepted Answer

Introduction / Context:With constant horizontal speed, times are proportional to horizontal distances. Angles of depression give those distances via tangent with the fixed tower height canceling out in ratios. Given Data / Assumptions: Depression changes 30° → 45° in 12 min. Motion along the line toward the tower; level ground; constant speed. Concept / Approach:Let tower height be H (cancels). Horizontal distances: d30 = H / tan 30° = H√3; d45 = H / tan 45° = H. The 12 minutes correspond to d30 − d45. From 45° to the tower is d45. Time scales with distance at constant speed. Step-by-Step Solution: Travel in 12 min: Δ = H√3 − H = H(√3 − 1).Speed v = Δ / 12 (arbitrary H cancels).Remaining distance from 45°: d45 = H.Time needed T = d45 / v = H / (H(√3 − 1)/12) = 12/(√3 − 1) = 6(√3 + 1).Numeric: √3 ≈ 1.732 → T ≈ 6 * 2.732 = 16.392 min ≈ 16 min 23 sec. Verification / Alternative check:Ratio method: T / 12 = d45 / (d30 − d45) = H / (H(√3 − 1)) → same value. Why Other Options Are Wrong:They correspond to rounding or formula mistakes; only ~16:23 matches exact evaluation of 6(√3 + 1) minutes. Common Pitfalls:Using depression distances incorrectly; forgetting to rationalize; rounding √3 too early leading to wrong seconds. Final Answer:16 min 23 sec.

Changing depression from 30° to 45°: From the top of a vertical tower, a car approaches in a straight line on level ground. The angle of depression changes from 30° to 45° in 12 minutes. How much additional time is required (after it is 45°) for the car to reach the foot of the tower?

More Questions from Height and Distance

Discussion & Comments