Suppose water is leaking from a tank through a circular hole of area An at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cApv 2gh, where c (0
Suppose water is leaking from a tank through a circular hole of area An at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cApv 2gh, where c (0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cAhy 2gh, where c (0 < c < 1) is an empirical constant.
A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (Assume the removed apex of the cone is of negligible height and volume.)
(a) Suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. The differential equation governing the height h in feet of water leaking from a tank after t seconds is
dh
5
dt
6h 3/2 .
In this model, friction and contraction of the water at the hole are taken into account with c = 0.6, and g is taken to be 32 ft/s. See the figure below.
8 ft
20 ft
circular hole
If the tank is initially full, how long will it take the tank to empty? (Round your answer to two decimal places.)
minutes
(b) Suppose the tank has a vertex angle of 60° and the circular hole has radius 3 inches. Determine the differential equation governing the height h of water. Use c = 0.6 and g = 32 ft/s<.
dh
dt
If the height of the water is initially 12 feet, how long will it take the tank to empty? (Round your answer to two decimal places.)
min
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