Water leaks out of a barrel so that the rate of change in the water level is proportional to the square root of the depth of the water at that time. if the water level starts at 30 inches and drops to 29 inches in 1 hour, how many hours will it take for all of the water to leak out of the barrel?

Let y(t) represent the level of water in inches at time t in hours. Then we are given ...
  y'(t) = k√(y(t)) . . . . for some proportionality constant k
  y(0) = 30
  y(1) = 29

We observe that a function of the form
  y(t) = a(t - b)²
will have a derivative that is proportional to y:
  y'(t) = 2a(t -b)

We can find the constants "a" and "b" from the given boundary conditions.
At t=0
  30 = a(0 -b)²
  a = 30/b²
At t=1
  29 = a(1 - b)²
  29 = 30(1 - b)²/b²
  29b² = 30b² -60b +30
  b² -60b +30 = 0
  (b -30)² = 870
  b = 30 +√870 . . . . . . b < 1 is an extraneous solution

The value of b is the time it takes for the height of water in the tank to become 0. It is 30+√870 hours ≈ 59 hours 29 minutes 45 seconds

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