Oompa Loompas have been running amok in Willy Wonka's Chocolate Factory, damag- ing one of the hot chocolate pipes. In the first hour, the leakage from the pipe worsens, as apparent from the table below. After an hour the Oompa Loompas manage to gradually mitigate the leakage using adhesive tape, until it reaches a constant rate of 1 gal/min after 100 minutes. Time elapsed since leakage started (min) Leakage (gal/min) 10 20 30 40 50 60 70 80 90 100 < 2 2.5 3.5 8 11 16 10 2 1 (a) Give an upper sum and lower sum estimate on the quantity of hot chocolate (in gallons) that leaked after an hour. (b) Give a midpoint sum estimate on the quantity of hot chocolate (in gallons) that leaked after 1 hour and 40 minutes (use n = 5 sub-intervals). (c) There are 750 gallons in total in the container the pipe is connected to. How much time will have elapsed from the incident till the container is depleted in the best case scenario and in the worst case scenario?

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.