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  3. 2 a a cylindrical tank with constant crosssectional area au...

Question: 2 a a cylindrical tank with constant crosssectional area au...

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2. (a) A cylindrical tank with constant cross-sectional area Au has a drain hole, with constant area An, at the bottom of the tank through which the water is draining out. If the height of water in the tank at time t is h(t) it was shown in class that 14y
where g 32 ft/s2 represents gravitational acceleration. If A 5.0 ft and Ax = 0.04 ft2 and the initial height of the water is h(0-16 ft, how long does it take the tank to drain completely? (b) In the next problem, instead of solving the differential equation we use it as a formula to determine some of the coefficients in the equation; this is an example of an inverse problem. Other famous inverse problems include ultrasound imaging and reflection seismology which is employed in oil exploration. It seems that, by placing marks on the tank corresponding to the water level at intervals of, say, one hour one should be able to create a reasonably accurate clock. The idea is distinctly not new. Such clocks, known as clepsydra (from the Greek κλέπτευ, to steal, and tap hydor, water) are believed to have appeared in China around 4000 BCE. The known versions all have the rather inelegant property that, as dh/dt is not coustant, the distances between successive hourly marks decreases with time. Your task here is to use the formula (2), with 4 ft or Water flow A A(h) now assumed to depend on h, to design a 12-hour clepsydra with the dimensions shown above and shaped like the surface obtained by rotating the curve z = g(y) around the y-axis. You are to determine a formula for the function g describing the shape of your clepsydra, together with the radius of the circular bottom hole, in order that the water level h h(t) will fall at the constant rate of 4 inches per hour (be careful with the units here).
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