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Time from A to B: \(\fracd60\) hours - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A drone follows a straight flight path defined by the parametric equations $ x = 2t + 1, y = -3t + 4, z = t $, where $ t $ is time in seconds. Find the point on this path that is closest to the weather station located at $ (5, -2, 3) $.

📌 To find the point on the line closest to a fixed point, we minimize the square of the distance between a general point on the line and the fixed point. Let the point on the line be $ (2t + 1, -3t + 4, t) $, and the fixed point be $ (5, -2, 3) $. The square of the distance is:

📌 To minimize $ D(t) $, take derivative and set to zero:

📌 ight) $ where $ C(t) $ reaches its minimum.

📌 ight) $), so the minimum occurs at the left endpoint:

📌 ight) $, so the minimum occurs as $ t o

📌 To find the exact value, approach from the right:

📌 ight) $, but the question asks for the value of $ t $ where this occurs. Since $ t =

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