Inception related rates

Calculus Level 5

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In the movie Inception, a bunch of people take a 10 hour flight in real life and sleep during the flight under a special sedative, putting them in a dream within a dream within a dream. So there are three dream levels:

  • Real time: 10 hour flight
  • Dream level 1: 10 hours x 20 = 200 hours pass in the 1st dream level
  • Dream level 2: 200 hours x 20 = 4000 hours pass in the 2nd dream level
  • Dream level 3: 4000 hours x 20 = 80000 hours pass in the 3rd dream level

In the third dream level, some guy decides to take a drink of water. So, at a constant rate, water is dispensed from a circular paraboloid-shaped water cooler into a cylindrical cup. The water cooler is \(y=x^2 \) revolved around the y-axis. The height of the cooler is 16 units, and the maximum radius of the cooler is 4 units. The cooler is filled to the top with water. The cup is 2 units tall, and has a radius of 1 unit.

It takes 1 second of second dream level time to fill the cup in the third dream level to the top using the water cooler. The water cooler dispenses water into the cup at a constant rate.

What is the rate of change of the height of the water level in the water cooler, in real time, at the point in time when the cup has just been filled to the top, in units/sec? If that rate equals \(R\), then find \( \left \lfloor 10^9 R \right \rfloor\).


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