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how many sides does a circle have - Belip

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Question: An oceanographer models the depth $ d(t) $ (in meters) of a marine layer influenced by tidal forces as a cubic polynomial in time $ t $ (in days). It is known that $ d(1) = 1 $, $ d(2) = 8 $, $ d(3) = 27 $, and $ d(4) = 64 $, and $ d(t) $ achieves its minimum depth exactly once. Find $ d(0) $.

📌 Solution: We are given that $ d(t) $ is a cubic polynomial satisfying:

📌 d(1) = 1,\quad d(2) = 8,\quad d(3) = 27,\quad d(4) = 64.

📌 = 1^3,\quad 8 = 2^3,\quad 27 = 3^3,\quad 64 = 4^3.

📌 So $ d(n) = n^3 $ for $ n = 1,2,3,4 $. Since $ d(t) $ is a cubic polynomial and agrees with $ t^3 $ at four distinct points, by uniqueness of interpolation,

📌 d(t) = t^3.

📌 But $ d(t) = t^3 $ is strictly convex and has no minimum (it decreases on $ (-\infty, 0) $, increases on $ (0, \infty) $), so it does not achieve a minimum. However, the problem states that the **minimum depth is achieved exactly once**, which implies $ d(t) $ has a unique global minimum. But $ t^3 $ has no such minimum. Contradiction?

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