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Eclair_de_XII
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Homework Statement
"Fluid will leak out of a hole at the base of a cylindrical container at a rate proportional to the square root of the height of the fluid's surface from the base. If a cylindrical container is initially filled to a height of ##h(0)=12 in.##, and it takes one minute for the height to reach ##h(60)=3 in.##, how long will it take for all of the fluid to leak out?"
Homework Equations
##h(0)=12 in.##
##h(60)=3 in.##
##\frac{dh}{dt}=-u(t)h(t)##
##u(t)=\sqrt{h}##
The Attempt at a Solution
##\frac{dh}{dt}=-u(t)h(t)=h^{\frac{3}{2}}##
##h^{-\frac{3}{2}}dh=-dt##
##\int h^{-\frac{3}{2}}dh=-\int dt##
##-2h^{-\frac{1}{2}}=-t+C##
##h^{-\frac{1}{2}}=2t+C##
##h(t)=\frac{1}{(2t+C)^2}##
I am very completely sure that this is not correct. For one, ##h(t)## can never be zero. I know I went wrong when trying to derive the expression for ##\frac{dh}{dt}##. And I have two conditions at ##t=0## and ##t=60## that I don't think will ever be satisfied simultaneously. What am I doing wrong?
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