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📰 Question: An underwater robot’s depth $ d(t) $ (in meters) satisfies $ d(t) = pt^3 + qt^2 + rt + s $. Given $ d(1) = 10 $, $ d'(1) = 12 $, $ d(2) = 28 $, and $ d'(2) = 30 $, find $ d(0) $. 📰 Solution: $ d(t) = pt^3 + qt^2 + rt + s $. Compute $ d'(t) = 3pt^2 + 2qt + r $. From $ d(1) = p + q + r + s = 10 $, $ d'(1) = 3p + 2q + r = 12 $, $ d(2) = 8p + 4q + 2r + s = 28 $, $ d'(2) = 12p + 4q + r = 30 $. Subtract first equation from third: $ 7p + 3q + r = 18 $. Subtract $ d'(1) $ from this: $ (7p + 3q + r) - (3p + 2q + r) = 4p + q = 6 $. From $ d'(2) $: $ 12p + 4q + r = 30 $, and $ d'(1) $: $ 3p + 2q + r = 12 $. Subtract: $ 9p + 2q = 18 $. Now solve $ 4p + q = 6 $ and $ 9p + 2q = 18 $. Multiply first by 2: $ 8p + 2q = 12 $. Subtract: $ p = 6 $. Then $ 4(6) + q = 6 $ → $ q = -18 $. From $ d'(1) $: $ 3(6) + 2(-18) + r = 12 $ → $ 18 - 36 + r = 12 $ → $ r = 30 $. From $ d(1) $: $ 6 - 18 + 30 + s = 10 $ → $ s = -8 $. Thus, $ d(0) = s = -8 $. Final answer: $ oxed{-8} $. 📰 Question: A philosopher of science considers a function $ k(x) $ modeling the "distance" from theory to observation, satisfying $ k(x + y) = k(x) + k(y) - 2k(xy) $. If $ k(1) = 1 $, find $ k(2) $. 📰 Discover The Secret Behind Salem 2 What Happened In The Town That Changed Everything 8793565 📰 Can Ucb Break Out The Surprising Surge You Dont Want To Missheres Why 5539665 📰 The Ultimate Carnivore Diet Food List That Will Transform Your Meals Overnight 1367966 📰 Perhaps The Problem Expects The Answer In Hours As A Decimal 1382205 📰 Download The Monkey Appits Smarter Than You Think 3864394 📰 Here Champaign Illinois 9820420 📰 Nutrition Facts Of Gala Apple 4660849 📰 Download This Life Changing Bitcoin Mining Applive Profits Guaranteed 7935563 📰 The Et Stock Frenzy Explained Rumors Risks And Rewards In The Hottest Debate 6597085 📰 Finally The Last Witch Hunter Film Series Ends But Is This The End Of The Witch Countdown 8459283 📰 Purdues Fletcher Loyer Claimed Rims Were Uneven At Baha Mar 8766899 📰 Base Trim Secrets Revealed These High End Details Are Changing Home Design Forever 919739 📰 Local Weekend Events 8089758 📰 Games Games Games Games Games Games Games 7614362 📰 Wake Up To A Tropical Twistif You Love Pina Coladas Try This 5154893