- #1
KillerZ
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Homework Statement
A tank in the form of a right-circular cylinder standing on end is leaking water through a circular hole in its bottom. When friction and contraction of water at the hole are ignored, the height h of the water in the tank is described by:
[tex]\frac{dh}{dt} = -\frac{A_{h}}{A_{w}}\sqrt{2gh}[/tex]
where Aw and Ah are the cross-sectional areas of the water and the hole, respectively.
a) Solve the DE if the inital height of the water is H. By hand, sketch the graph of h(t) and give its interval I of definition in terms of the symbols Aw. Ah, and H. Use g = 32 ft/s^2.
b) Suppose the tank is 10 feet high and has radius 2 feet and circular hole has radius 1/2 inch. If the tank is initially full, how long will it take to empty?
Homework Equations
[tex]\frac{dh}{dt} = -\frac{A_{h}}{A_{w}}\sqrt{2gh}[/tex]
The Attempt at a Solution
I want to make sure I did this part right before attempting b) as it needs the answer for a).
a)
[tex]\frac{dh}{dt} = -\frac{A_{h}}{A_{w}}\sqrt{2gh}[/tex]
[tex]\frac{dh}{dt} = -\frac{A_{h}}{A_{w}}\sqrt{64H}[/tex]
[tex]\frac{dh}{\sqrt{64H}} = -\frac{A_{h}}{A_{w}}dt[/tex]
[tex]\int\frac{dh}{\sqrt{64H}} = -\int\frac{A_{h}}{A_{w}}dt[/tex]
[tex]\frac{h}{\sqrt{64H}} = -\frac{A_{h}}{A_{w}}t[/tex]