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Understanding the Context

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Cientista John Calhoun estudou o impacto da superpopulao em ratos no experimento "Universo 25". Em ambiente com recursos ilimitados, a populao de ratos cresceu at apresentar.

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The Undertaker Ex-Wife Sara Calaway Age, Family & Bio 2021

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The Undertaker Ex-Wife Sara Calaway Age, Family & Bio 2021
Tagged Sara Calaway - FamousFix
Tagged Sara Calaway - FamousFix
Tagged Sara Calaway - FamousFix
Tagged Sara Calaway - FamousFix

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📰 Solution: Assume $ V(t) = at^2 + bt + c $. From $ V(1) = a + b + c = 120 $, $ V(2) = 4a + 2b + c = 200 $, $ V(3) = 9a + 3b + c = 300 $. Subtract first equation from the second: $ 3a + b = 80 $. Subtract second from the third: $ 5a + b = 100 $. Subtract these: $ 2a = 20 $ → $ a = 10 $. Then $ 3(10) + b = 80 $ → $ b = 50 $. From $ a + b + c = 120 $: $ 10 + 50 + c = 120 $ → $ c = 60 $. Thus, $ V(t) = 10t^2 + 50t + 60 $. For $ t = 4 $: $ V(4) = 10(16) + 50(4) + 60 = 160 + 200 + 60 = 420 $. Final answer: $ oxed{420} $. 📰 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} $.