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📅 April 30, 2026 👤 scraface

Apr 30, 2026

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📌 E(1) = E(0) + 1 = 1, \quad E(2) = 1 + (2\cdot1 + 1) = 1 + 3 = 4, \quad \text{etc.}

📌 So $ E(n) = \sum_{k=0}^{n-1} (2k + 1) = n^2 $, since sum of first $ n $ odd numbers is $ n^2 $.

📌 Hence, $ E(10) = 10^2 = 100 $.

📌 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.

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