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Water drains from a vertical cylindrical tank through a valve at the base. The rate at which the water level falls is proportional to the square root of the water depth y, with positive constant k. With time t in minutes and k = 1/15, how long does it take to empty the tank if the water is initially 4 m deep?
- 30 min
- 45 min
- 60 min
- 80 min
Correct answer: 60 min
Solution
Solving the separable ODE gives 2*sqrt(y) = -k*t + C; using y = 4 at t = 0 fixes C, and setting y = 0 gives t = 2*sqrt(4)/k = 4/k = 60 min.
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