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# Differential Equation : Torricelli's Law

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(1) (a) Let h(t) and V (t) be the height and volume of water in a cylindrical tank at time t. If water leaks through a circular hole with area a at the bottom of the tank, Torricelli's law says that the rate of change of volume is given by the equation
dV
--- = - a*sqrt(2gh)
dt
where g is the acceleration due to gravity.

(i) Show that the height of the water decreases according to the equation
dh
-- = = - alpha*sqrt(h)
dt

for some constant alpha. What is alpha?

(ii) If we let h(t) = u(t)^2, find a differential equation for u.

(iii) Given that the tank is initially full and the height of the tank is 6 metres and the radius of the tank is 2 metres, how long does it take for all the water to drain from the tank, if the hole has radius 1cm?