Find the time required for a cylindrical tank of radius 8 ft.and height 10 ft. t

Question

Find the time required for a cylindrical tank of radius 8 ft.and height 10 ft. to empty through a round hole of radius 1 in atthe bottom of the tank, given that water will issue from such ahole with velocity approximately v = 5h ft/s, where h is thedepth of the water.
please show steps to solve Find the time required for a cylindrical tank of radius 8 ft.and height 10 ft. to empty through a round hole of radius 1 in atthe bottom of the tank, given that water will issue from such ahole with velocity approximately v = 5h ft/s, where h is thedepth of the water.
please show steps to solve

Explanation / Answer

Let height of water at some point of time be h. Then velocity of water coming out = 5h ft /s In time dt, a volume dV = 5hdt* [*(1/12)2] of water will comeout. So in this time, the height will decrease by dV/A where A isthe cross-sectional area = 82 ft2 =64 => dh = -dV/A    [ The minus sign becauseheight is decreaing with progressing time] => dh = - 5h dt / 144*64 Solving by seperation of variables => dh/h = -(5/144*64) dt => 2h |0H = -(5/144*64)t |T0   where H is the height and Tis the time taken to empty => -2H = -(5/144*64) T => T = 210*64*144 /5 s = 11657s =3hr 14 min 17s => dh = -dV/A    [ The minus sign becauseheight is decreaing with progressing time] => dh = - 5h dt / 144*64 Solving by seperation of variables => dh/h = -(5/144*64) dt => 2h |0H = -(5/144*64)t |T0   where H is the height and Tis the time taken to empty => -2H = -(5/144*64) T => T = 210*64*144 /5 s = 11657s =3hr 14 min 17s

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